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1 . Question

What percentage does 90 represent in relation to 360?

2 . Question

7 added to 4 times a quantity y is equal to 62. Which equation represents this condition?

• a. 7(4y) = 62
• b. 7 – 4y = 62
• c. 7 + 4y = 62
• d. 4y – 7 = 62

3 . Question

For a cost of $57, how many pounds of avocados were bought at a rate of $19 per pound?

4 . Question

The table displays three sets of values for x and their corresponding f(x) values for a linear function f. What equation accurately represents the definition of f(x)?

• a. f(x) = 6x + 65
• b. f(x) = 8x + 65
• c. f(x) = 65x + 73
• d. f(x) = 73x + 81

5 . Question

If \frac{x}{15}=20 , what is the value of \frac{15}{x} ?

6 . Question

What is the equation that defines line p in the xy-plane, given that it passes through the point (-4, 6) and has a slope of 6?

• a. y = 6x + 30
• b. y = 6x – 30
• c. y = 6 + 30x
• d. y = 6x – 30x

7 . Question

2.5b + 5r = 80

The provided equation expresses the connection between the quantity of biscuits (b) and raisins (r) that a reviewer business can handle in a single day. If the business is tending to 16 raisins on a specific day, what is the capacity for biscuits that it can manage on the same day?

8 . Question

In a specific rectangular area, the length-to-width ratio is 45 : 15. If the width of the region grows by 8 units, how should the length be adjusted to uphold this ratio?

• a. It must decrease by 24.5 units.
• b. It must increase by 24.5 units.
• c. It must decrease by 24 units.
• d. It must increase by 24 units.

9 . Question

In a circle with center O, the arc XY measures 105°. What is the degree measure of the angle XOY that corresponds to this arc?

10 . Question

In a right triangle, the side lengths are 3\sqrt3 , 2\sqrt3 ,and \sqrt{70} units. What is the area of the triangle in square units?

*Enter in just the number for your answer.

All SAT Practice Tests

If you want some more in-depth prep, use a free SAT practice test listed below.

• Practice Exams = Timed and Full-Length
• Practice Sets = Not Timed and Smaller Sets of Questions

SAT Practice Exam #1

SAT Practice Exam #2

SAT Practice Exam #3

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SAT Math Practice Sets

SAT Reading & Writing Practice Sets

If you want some additional help with prepping for the SAT, consider using  SAT prep courses .

Official SAT PDF Practice Exams

If you want to study using PDFs, use the links below. These are full-length practice exams provided by the CollegeBoard.

Overview of the SAT

The SAT is an entrance exam used by colleges and universities to help make decisions about admissions.

The SAT is administered by the CollegeBoard and is given 7 times per year. The exam is a timed, mainly multiple-choice exam, taken by students in high school.

Beginning in the spring of 2024, the SAT will be going fully digital. The digital SAT (DSAT) will have some formatting and content changes to it, along with some other changes. You can review those changes below.

New SAT (Digital SAT)

This exam will be given to students starting in spring of 2024.

Old SAT (Written Exam)

This exam will be given to students up until spring of 2024.

The biggest difference between the old SAT and the digital SAT is that the digital SAT is taken completely online and has combined some sections to streamline the exam.

Scoring of the SAT

You will receive a score for math and a score for reading/writing. Each of those scores will be between 200 and 800 .

Your total score will be the sum of those two scores. You can receive a total score between 400 and 1600 on the SAT.

The average SAT score is 1060. Learn more about good SAT scores .

Registering for the SAT and Test Dates

You can register for the SAT online via the CollegeBoard site .

You will need to do the following when registering for the SAT:

• Have a Valid Photo ID
• Upload a Photo of Yourself When Registering Online
• Pay Registration Fees
• Print Your Admission Ticket

It will cost your $60 to take the SAT. There may be some additional fees like canceling, changing test center, etc..

The SAT is typically given 7 times per year. The exam is given on Saturdays. You can expect the exam to be given in these months:

• Early March
• Late August
• Early October
• Early November
• Early December

You can find exact dates on the official website .

Steps for Using SAT Practice

When using our free SAT practice tests, we recommend the following steps to get the most out of your time:

• Take 1 practice exam from each subject.
• Determine which subject you struggled the most with.
• Focus on that 1 subject moving forward. Take additional practice exams in that subject, study important concepts, and invest in a SAT prep course if need be.
• Take a practice exam on that subject again and see where you stand. Continue to focus on that subject if you need more help or move on to another subject if you did well on this practice exam.
• Repeat steps 3 and 4 for other subjects.

Benefits of Using SAT Practice Questions

There are many benefits to using SAT questions during your prep process. Some of those benefits include:

Help With Timing

The SAT exam is a timed test. Keeping a steady pace is critical to achieving a high score.

You can improve your decision making and your time by taking practice exams.

Test Familiarity

All standardized tests, including the SAT, have their own unique way of presenting questions and answer choices.

You will gain more familiarity and comfort with the SAT question style as you take more practice quizzes. On the real exam day, there will be no surprises.

Efficient Studying

When you take many practice exams, you will get a sense of your test strengths and weaknesses.

Many students mistakenly spend time working on their strengths while ignoring their weaknesses.

Knowing which subjects you struggle with will help you focus your study time.

Work On Problem Solving

Tests like the SAT measure your ability to solve problems, not just memorize information. It is critical to have strong problem-solving abilities.

The answer explanations provided in our score reports can help you understand how to solve problems that you may be struggling with.

Frequently Asked Questions

When is the sat going to the digital format.

The SAT will be switching to the digital format in the spring of 2024.

How many questions are on the SAT?

There are 154 questions on the old SAT (written version).

There are 98 questions on the new SAT (digital version).

How much time do you have to take the SAT?

You will have 3 hours and 15 minutes to take the old SAT (written version).

You will have 2 hours and 14 minutes to take the new SAT (digital version).

Can I use a calculator on the math section of the SAT?

There will be 1 math section in which you can use a calculator and 1 math section in which you cannot use a calculator on the old SAT (written version).

You will be permitted to use a calculator on all math modules on the new SAT (digital version).

What is a good way to practice for the SAT?

We recommend taking 1 practice exam for each subject. You can then determine which subject gave you the most trouble and focus your studies on that subject.

• Authored By: Adam Groden
• Last Updated: January 19, 2024

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SAT Prep Courses & Tutoring

Free SAT Practice Questions

Doesn't everyone love the word "free"? This could not be truer—especially in the world of test prep. Practice makes perfect—or at least that's how the old saying goes—and free SAT practice means the world to ambitious students with high-scoring goals. If you're looking for a thorough breakdown of how to solve particular SAT question types, then we've got the free practice that could be a real game-changer in your course of study. Our step-by-step explanations illustrate for you what to expect from what each SAT question, revealing question-specific hurdles and common test-related traps.

Each of our free 60 practice questions is accompanied with a detailed explanation to clarify why a particular answer is correct, but even taking it one step further and entailing why the others are incorrect. We believe a well-rounded approach to understanding the SAT is crucial for success, which is why our breakdowns of important sample questions facilitate more than one way to arrive at the right choice. In the end, the only way to find out your trouble spots and successes is to practice, practice, practice—so why not get cracking on what we have to offer so you can ensure your destiny with a high score.

You have not answered any question so far. You can answer all questions in a row (click on "All Questions") or only all questions of a particular section (click on that Section) or a single selected question (click on that Question).

SAT is a registered trademark of the College Board, which is unaffiliated with and does not endorse this website.

Help your child succeed in their SATs

The sats paper guide - free advice and help for the sats.

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Maths Reasoning Worksheet and Explanation

We’ve structured this information to help children with their education. It is targeted at children in years 5 and 6 and the questions for the worksheet have been stripped from past papers. Having an understanding of maths reasoning is part of the primary school curriculum and children will deal with maths reasoning in both KS1 and KS2.

The Information below will give an overview of the topic and we have included a detailed worksheet with full answers. The worksheet includes 30 maths reasoning questions and is relevant for KS2 pupils approaching their SATS test.

Download Free Maths Reasoning Worksheet – Questions

Download free maths reasoning worksheet – answers.

If you are not ready to download the worksheets yet, then read on for some information about maths reasoning. This has provided to introduce the topics covered in the worksheet for those that might be unfamiliar but also as a quick revision tool for those that would like a quick refresher before accessing the worksheet.

Maths Reasoning Explained

The purpose of maths reasoning.

Reasoning allows us to make sense of the world around is and problems we might face on a day-to-day basis. Maths reasoning is part of this ability and helps us to use maths in meaningful ways and in applied settings. The question sheet uses examples that we might come across in everyday life where we need to use our maths skills to make an informed decision, such as the correct train to catch or how much money I will have remaining after I have been to the shops.

How do I Develop my Maths Reasoning?

Maths reasoning is dependent on the core skills learnt in the primary curriculum and is essentially utilising these in applied scenarios. In order to improve your maths reasoning skills, it is therefore important that you continue to develop your core skills. It is also important that students are able to explain their maths reasoning using the correct terminology wherever possible.

Where Maths Reasoning is Used?

As mentioned above reasoning is something we use every day, usually without thinking. These groups describe most of the situations where we will most likely be using our maths reasoning skills:

• The first encounter of a new challenge
• When a range of starting locations are possible
• When there is missing information
• When selecting a problem-solving skill
• When evaluating a solution in context
• When there is more than one solution
• When logical and critical thinking are required
• When there are multiple ways of solving a problem
• The basis for mathematical proof

Please feel free to download the maths reasoning worksheet and answers. No registration is necessary.

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SATs Resources and Worksheets

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Assessments written with SATs style questions and papers for primary teachers of Year 6 children. Fluency, reasoning and problem solving questions for maths, as well as arithmetic papers . All SATs subjects and question styles are covered, including grammar, punctuation and spelling papers , and reading . SATs questions are mapped to the SATs sample paper, as well as past papers, to give your children the opportunity to practice specific question styles in bulk.

Arithmetic SATs Practice Resources & Worksheets

Grammar, punctuation and spelling sats practice resources & worksheets, key stage 2 boost timetable, reading sats practice resources & worksheets, reasoning sats practice resources & worksheets.

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Year 2 Change the Sentence Types Test Practice

Change the Sentence Types Year 2 worksheets based on the 2015 KS2 Grammar, Punctuation and Spelling (Levels 3 – 5) English test, question 36, in 3 levels.

KS2 Boost timetable

In order to save you time, our weekly Boost timetable covers the domains for maths and GPS.

2019 KS2 SATs Arithmetic Assessment Pack 5

This 2019 KS2 SATs Arithmetic Assessment Pack contains five different assessment papers based on questions from the 2019 key stage 2 arithmetic assessment along with an accompanying Teacher Pack.

2019 KS2 SATs Arithmetic Assessment Pack 4

2019 KS2 SATs Arithmetic Assessment Pack 3

2019 KS2 SATs Arithmetic Assessment Pack 2

2019 KS2 SATs Arithmetic Assessment Pack 1

2022 KS2 SATs Arithmetic Assessment Pack 5

This 2022 KS2 SATs Arithmetic Assessment Pack contains five different assessment papers based on questions from the 2022 key stage 2 arithmetic assessment along with an accompanying Teacher Pack.

2022 KS2 SATs Arithmetic Assessment Pack 4

2022 KS2 SATs Arithmetic Assessment Pack 3

2022 KS2 SATs Arithmetic Assessment Pack 2

2022 KS2 SATs Arithmetic Assessment Pack 1

Explain the Meaning of the Modal Verb KS2 SATs GPS Test Practice

Explain the Meaning of the Modal Verb KS2 SATs Test Practice 2018 questions have been taken from the KS2 GPS test to help your children practise specific question types.

Which Sentence is Written in Standard English? KS2 SATs GPS Test Practice

Which Sentence is Written in Standard English? KS2 SATs Test Practice 2016, 2017 and 2018 questions have been taken from the Sample KS2 Grammar, Punctuation and Spelling test to help your children practise using Standard English.

Numbers in Digits and Words SATs KS1 Reasoning Test Practice

These KS1 Reasoning Numbers in Digits and Words questions have been taken from the 2016 Sample KS1 Reasoning test to help your children practise specific question types.

Identifying Given Word Classes in Sentences KS2 SATs GPS Test Practice

Identifying Given Word Classes in Sentences KS2 SATs Test Practice 2018 questions have been taken from the Sample KS2 Grammar, Punctuation and Spelling test to help your children practise using Word Classes.

Tick the Incorrect Punctuation KS2 SATs GPS Test Practice

Tick the Incorrect Punctuation KS2 SATs Test Practice questions have been taken from the KS2 GPS test to help your children practise specific question types.

KS1 Arithmetic Subtracting Multiples of 10 SATs Test Practice

These KS1 Arithmetic subtracting multiples of 10 questions have been taken from the KS1 Arithmetic test to help your children practise specific question types. Questions of this style also appear on the 2016 sample Arithmetic paper (question 11), the 2017 Arithmetic Test (questions 8 and 20), the 2018 Arithmetic Test (questions 10 and 17) and the 2019 Arithmetic Test (questions 5 and 15).

KS1 Arithmetic Find the Missing Number Subtraction Sentences SATs Test Practice

These KS1 Arithmetic Find the Missing Number Subtraction Sentences questions have been taken from the 2016 sample Arithmetic paper (question 12), the 2016 Arithmetic Test (questions 3 and 9), the 2017 Arithmetic Test (question 23), the 2018 Arithmetic Test (question 23) and the 2019 Arithmetic Test (question 22) Arithmetic test to help your children practise specific question types.

Rounding KS2 Reasoning Test Practice

Five differentiated activities for rounding numbers to practise question types from the KS2 Sample Reasoning Paper.

Multiplying 2, 3 and 4 Digit Numbers by 2 Digit Numbers KS2 Arithmetic SATs Test Practice

These KS2 Arithmetic Multiplying 2, 3 and 4 Digit Numbers by 2 Digit Numbers questions have been taken from the 2018 KS2 Arithmetic, 2017 KS2 Arithmetic, 2016 KS2 Arithmetic and Sample 2016 KS2 Arithmetic test to help your children practise specific question types.

Change the Conjunction KS1 GPS SATs Test Practice

Change the Conjunction KS1 SATs Test Practice questions have been taken from the Sample KS1 Grammar, Punctuation and Spelling test to help your children practise using specific question types.

KS2 Arithmetic Answer First Subtraction SATs Test Practice

These KS2 Arithmetic Answer First Subtraction questions have been taken from the 2018 KS2 Arithmetic test to help your children practise specific question types.

KS2 Arithmetic Answer First Subtracting Multiples of 10, 100 and 1,000 SATs Test Practice

These KS2 Arithmetic Answer First Subtracting Multiples of 10, 100 and 1,000 questions have been taken from the 2016 KS2 Arithmetic test to help your children practise specific question types.

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SAT Math Problems with Solutions

If you are in the middle of your SAT studying, then I’m sure you understand the importance of practicing SAT math problems as you learn the various SAT math topics. In this article, we will provide some math background and present various SAT math practice problems and answers.

The problems presented here are just a start, but you can get additional practice by checking out the Target Test Prep online SAT course .

Here are the topics we’ll cover in this article:

An overview of the sat math section, the ttp and sat breakdown of sat math topics, sat math subtopics, using your calculator on the sat, the two major question types on the sat, multiple-choice question 1, multiple-choice question 2, multiple-choice question 3, multiple-choice question 4, multiple-choice question 5, multiple-choice question 6, multiple-choice question 7.

• Grid-In Question 1

Grid-In Question 2

Grid-in question 3, grid-in question 4, grid-in question 5, grid-in question 6, grid-in question 7.

• Don’t Try to Predict Which SAT Math Topics You’ll See

Preparing for SAT Math

Topical learning with ttp, what math problems are on the sat, is the math on the sat hard, what is the hardest math subject in sat tests, what’s next.

Before we jump into SAT math practice questions, let’s discuss some basics of SAT math.

There are four sections on the SAT, and SAT math is tested in section 3 and section 4. Section 3 presents 20 questions, which you have 25 minutes to answer. Section 4 contains 38 questions, which you have 55 minutes to answer.

Furthermore, section 3 does not permit the use of a calculator, while section 4 does.

Additionally, each section contains both multiple-choice and grid-in questions, both of which we’ll discuss shortly.

Sections 3 and 4 of the SAT are math and are 25 minutes and 55 minutes long, respectively.

Next, let’s discuss the SAT math topics that could show up on your exam.

SAT Math Topics

So, here is some good news! Many math topics that can show up on your SAT are topics that you have already studied or will study soon in your high school math curriculum. So, as you jump into your SAT math prep, some topics should not be new to you, at least on a foundational level. And don’t worry, if some are new, you can learn them!

Many SAT math topics should be familiar to you.

Now, let’s discuss the breakdown of the math concepts tested on the SAT.

It’s important to note that although the College Board has its way of categorizing SAT math topics, we at TTP break them down a bit more granularly. For now, we can start with the College Board’s categorization, which includes four major SAT math categories:

• Heart of Algebra
• Problem Solving and Data Analysis
• Passport to Advanced Math
• Additional Topics in Math

Let’s subdivide these categories, the “TTP way,” so you can get a better overview of what is tested.

Heart of Algebra:

• Solving Linear Equations
• Inequalities and Absolute Value
• Coordinate Geometry
• Linear Functions
• Systems of Linear Equations

Problem Solving and Data Analysis:

• General Word Problems
• Unit Conversions
• Percentages

Passport to Advanced Math :

• Quadratic Equations
• Graph Interpretation

Additional Topics in Math :

• Trigonometry
• Complex Numbers

So, among the College Board’s four major categories, there are 22 math topics tested. However, keep in mind that these are the main topics. Each main topic branches into hundreds of subtopics, so let’s discuss that now.

There are 22 major SAT math topics.

You may think that studying just 22 math topics is a piece of cake. However, you can’t forget that there are hundreds of subtopics! For example, consider the major math topic of Linear Equations. Branching off from that major topic are smaller subtopics such as:

• Solving an Equation for One Variable
• Solving a System of Equations for Two Variables
• The Combination Method
• Solving for One Variable in Terms of Other Variables
• Equations with Fractions
• When a System of Linear Equations Has No Solution

In TTP, the chapter on Linear Equations actually has 23 subtopics! So, when you’re prepping for the SAT math test, just make sure to dig deep into the major topics and come out on the other side having also mastered the subtopics. This approach to your prep will make you an absolute force on test day!

When studying each major SAT math topic, learn the subtopics too.

Before jumping into our discussion and practice of the two major question types in SAT math, with answer explanations, let’s discuss calculator use on the SAT.

We have already discussed that you can use your calculator only in section 4 of the SAT. For all other sections, it must be put away and out of sight.

It’s also important to understand that your calculator can be a tool that can either work to your advantage or hurt you, depending on how and when you are using it.

For instance, if you find that you start using your calculator to do simple math calculations, then you likely are wasting time. For example, you should not need your calculator to calculate the sum of 11 and 13 or the product of 10 and 12, right? Conversely, you do want to use your calculator when determining the product of 3.56 and 11.2 or when dividing 0.348 by 0.12.

Certainly, you have to use your judgment, but getting a feel for when to use the calculator and when not to will be huge for test day. Regarding which calculator to use, check out the list from College Board to see the exact calculators allowed for the SAT. Our recommendation is to use either a graphing calculator or a scientific calculator, as those have more functionality than a basic 4-function calculator.

No matter which calculator you choose, make sure you practice using it well before test day, so you do not have any unnecessary issues during your SAT.

Ensure that you are strategically using your calculator on the SAT.

Now, let’s discuss the two major SAT question types.

SAT math questions are of two types: multiple-choice and grid-in. Of the 58 math questions on the SAT, you will encounter 45 multiple-choice and 13 grid-in questions.

If we break down the number by each section, section 3 (no calculator) contains 15 multiple-choice questions and 5 grid-in questions, while section 4 (calculator) contains 30 multiple-choice and 8 grid-in questions.

There are 45 multiple-choice and 13 grid-in questions on the SAT.

Next, let’s discuss multiple-choice questions in further detail and practice some questions!

Multiple-Choice Questions

Multiple-choice questions on the SAT are pretty much what you would expect, with one small difference. In a traditional multiple-choice question, you have five answer choices (A, B, C, D, and E). However, in SAT multiple-choice questions, you have just four answer choices (A, B, C, and D).

There are four answer choices in multiple-choice questions on the SAT.

Now, let’s practice some SAT multiple-choice questions.

SAT Major Topic: Passport to Advanced Math

TTP Major Topic: Quadratic Equations

TTP Subtopic: The Discriminant

x 2 + 2 = 4x has how many real solutions?

• More than 2

It isn’t necessary to solve this equation for x. Note that we need only determine the number of real solutions. Thus, we can use discriminant analysis to efficiently answer the question.

First, let’s express the given equation in general form:

x 2 – 4x + 2 = 0

So a = 1, b = -4, and c = 2. Thus, for this equation, the discriminant is:

b 2 – 4ac = (-4) 2 – 4(1)(2) = 16 – 8 = 8

Because the discriminant’s value is positive, we know there are two real roots to the quadratic equation.

Problem Solving Question 2

SAT Major Topic: Heart of Algebra

TTP Major Topic: Arithmetic Word Problems

TTP Subtopic: Inequality Word Problems

A particular department store has 50 displays, and the total number of displays in the store cannot exceed 90. If no old displays are taken down and new ones are added at a rate of 3 per day, and if y represents time in days, which of the following inequalities represents the situation?

50 + 3y ≤ 90

• 50 – 3y ≤ 90

First, we should note that the situation described has all of the components of a linear inequality: There is an initial value (50 displays), a total capacity (90 displays), and a constant rate of increase (3 displays per day).

Both sides of the inequality will represent the number of displays:

Next, we can fill in the given numbers accordingly:

Finally, when we assemble these into the final inequality, we get:

SAT Major Topic: Problem Solving and Data Analysis

TTP Major Topic: Statistics

TTP Subtopic: Solving for an Unknown in an Average Equation

Tabitha swims the 200-meter freestyle three times. Her completion times for the first two swims are 30 seconds and 36 seconds. What is the completion time for her third swim?

We know that Tabitha’s average is 32 seconds. Her first two times are 30 seconds and 36 seconds. Let’s let x = the time of her third swim. So, using the formula for the average, we have:

average = sum/number

⇒ 32 = (30 + 32 + x)/3

⇒ 96 = 62 + x

TTP Major Topic: Functions

TTP Subtopic: Domain of a Function

Which of the following describes the domain of the function v(x) = x^5 + x^3?

• 3 < x < 5.
• The domain is all real numbers.

First, recall that the domain of a function specifies all allowable x values. Generally, for SAT purposes, we have domain restrictions when we have (1) a denominator that is equal to 0 or (2) a function for which we would take the square root of a negative number.

For example, if we had f(x) = 2/(x – 4), we would have a domain restriction of x = 4 because that value would make the denominator of the fraction equal to 0. A second example would be sqrt(x – 5). If x were any number less than 5, then we would be taking the square root of a negative number.

Clearly, the domain of the function v(x) has no restrictions, since we can take any real number to the fifth power or the third power, and we can add these two quantities for any real value of x. Thus, the domain of v(x) is all real numbers.

TTP Major Topic: General Word Problems

TTP Subtopic: Age problems

Harold is 30 years older than Paloma. If in 10 years, Harold will be 3 times as old as Paloma will be then, how old will Harold be in 3 years?

First, let’s define two variables:

H = Harold’s age today

P = Paloma’s age today

Next, we can create two equations from the information presented in the problem stem.

Since Harold is 30 years older than Paloma, we have:

⇒ H = P + 30

In 10 years, Harold will be (H + 10) years old, and Paloma will be (P + 10) years old. Thus, at that time, Harold will be 3 times as old as Paloma, and we have:

⇒ H + 10 = 3(P + 10)

⇒ H + 10 = 3P + 30

⇒ H = 3P + 20

Next, we can substitute P + 30 for H in the second equation:

⇒ P + 30 = 3P + 20

Therefore, Harold is currently 5 + 30 = 35 years old, so in 3 years, he will be 38 years old.

SAT Major Topic: Additional Topics

TTP Major Topic: Complex Numbers

TTP Subtopic: Combining Complex Numbers

What is the sum of the complex numbers 3 + 2i and 5 – 7i, where i = ?

First, we can add the real parts of 3 + 2i and 5 – 7i (3 and 5, respectively):

⇒ 3 + 5 = 8

Now, we can add the imaginary parts of 3 + 2i and 5 – 7i. Note that because 5 – 7i has subtraction between its two parts, the imaginary part of this complex number is -7. (We can perhaps see this more clearly if we rewrite 5 – 7i as the equivalent expression 5 + (-7i).) The imaginary part of 3 + 2i is 2:

⇒ 2i + (-7i) = -5i

Thus, the sum of 3 + 2i and 5 – 7i is 8 + (-5i), or, equivalently, 8 – 5i.

TTP Major Topic: Trigonometry

TTP Subtopic: Radian Measure

What is the equivalent of 72° in radians?

To convert 72˚ to radians, we can create the following proportion:

𝞹 radians / 180 degrees = x / 72 degrees

Next, we can remove the units and cross-multiply to solve for x:

⇒ 72𝞹 = 180x

⇒ 72𝞹 / 180 = x

⇒ 2𝞹 / 5 = x

Now, let’s discuss SAT grid-in questions.

Grid-In Questions

Technically, the make-up of grid-in questions differs from that of multiple-choice questions. A grid-in question will always have a numerical answer with no variables in the answer. It’s important to note, too, that some grid-in questions have multiple possible answers. You need to grid in only one of these answers.

Despite those differences, the skills needed to answer a grid-in question do not differ from those required to answer a multiple-choice question. The primary difference between multiple-choice questions and grid-in questions is that there are no answer choices to select from in a grid-in question.

There are 8 grid-in calculator questions and 5 grid-in non-calculator questions. Also, grid-in questions are always presented at the end of a section.

There are 8 grid-in calculator questions and 5 grid-in non-calculator questions.

An example of a blank grid is shown below:

There are a few rules to learn about filling in the grid:

• You must bubble in your answer. Writing your answer in the boxes at the top of each column is insufficient.
• Only positive numbers can be bubbled. Thus, if your answer is a negative number, you have made an error in your calculations.
• If your answer is a decimal number, round it to 3 decimal places, if necessary, and make sure you bubble one position for your decimal point. A leading zero is not required for decimal numbers between 0 and 1.
• Fractions need not be reduced. They can be entered in traditional fraction fashion, with the “slash” mark bubbled to separate the numerator and the denominator. Alternatively, you can express your fractional answer as a decimal number.

Now, let’s practice with a couple of grid-in examples.

TTP Major Topic: Exponents

TTP Subtopic: Multiplication of Exponents with the Same Base

If a and b are positive integers and 3 12 = 3 a 3 b , what is one possible value of a x b?

Since 3 a 3 b = 3 a+b , we can rewrite the given equation as 3 12 = 3 a+b . Since our base is 3 on both sides of the equation, a + b must be equal to 12. We need to determine the two values whose sum is 12. Because a and b must be positive integers, there are only a handful of options for their values: {1, 11}, {2, 10}, {3, 9}, {4, 8}, {5, 7}, and {6, 6}. The products of these pairs are: 11, 20, 27, 32, 35, 36.

Therefore, any of the values 11, 20, 27, 32, 35, or 36 are correct.

Answer: 11, 20, 27, 32, 35, or 36

Remember that sometimes there are multiple correct answers to a grid-in question. You need to bubble only one of them into the grid to get the question correct.

Let’s try another question.

TTP Major Topic: Arithmetic

TTP Subtopic: Fractions

Alvin accidentally spilled his marble collection on the floor. If he was able to recover 24 of the original 120 marbles, what fraction of his marble collection was he able to recover?

The problem gives us these two pieces of information:

Number of recovered marbles = 24

Total number of marbles = 120

Thus, we know that the fraction of recovered marbles is 24/120.

Because this is a grid-in question, we cannot directly grid in the value 24/128 because there are not enough positions in the grid to accommodate this fraction. Thus, we have two options: (1) reduce the fraction or (2) convert the fraction to a decimal number.

• Reduce the fraction:

Since both 24 and 120 are divisible by 24, we can divide the numerator and denominator by 24:

⇒ 24/120 = 1/5

• We can convert the fraction 24/120 to a decimal value:

⇒ 24/120 = 0.2

Answer: 1/5 or 0.2

We could bubble either 1/5 or 0.2 into the grid.

TTP Major Topic: Linear Equations

TTP Subtopic: Solving For Expressions

If 15x + 24y + 10 = 15, then what is the value of 5x + 8y?

To start, we can move 10 from the left side of the equation to the right side:

⇒ 15x + 24y = 5

Next, we see that each term in the equation on the left is a multiple of 3. Thus, if we divide the entire equation by 3, we have:

⇒ 5x + 8y = 5/3

Answer: 5/3

TTP Subtopic: Compound Inequalities

If n is an integer and 10 < 2n < 18, then what could be a value of n?

First, we need to isolate n. To do so, we can divide the entire inequality by 2, and we have:

⇒ 5 < n < 9

Since n is an integer, we see that n can be 6, 7, or 8. Any of these answers is correct.

Answer: 6, 7, 8

TTP Major Topic: Rate Questions

TTP Subtopic: Rate Questions Not Involving Distance

A tank is filled with 12,000 gallons of water. If a drain is opened that allows water to drain from the pool at a constant rate of 240 gallons per hour, how many gallons are drained from the pool after 6 hours?

We are given that a pool is filled with 12,000 gallons of water and that, when a drain is opened, water drains at a rate of 240 gallons per hour. So, to determine how much water has drained after 6 hours, we can use the following form of the general rate formula:

⇒ rate x time = amount

⇒ 240 x 6 = 1,440 gallons

Answer: 1,440 (Note: you will grid in 1440 with no comma)

TTP Subtopic: Symbolism Questions

If x@y = x 2 + 2y, what is 5@3?

The function statement x@y = x 2 + 2y sets up the instruction for evaluating the function. The second function statement tells us that we are to evaluate the function when x = 5 and y = 3.

When x = 5 and y = 3, we substitute these values into the function instruction, which is x 2 + 2y:

⇒ 5 2 + 2(3) = 25 + 6 = 31

SAT Major Topic: Heart of Algebra:

TTP Major Topic: Coordinate Geometry

TTP Subtopic: The Y-Intercept

In the xy-plane, the equation of line k is given as 5x + (2/3)y = 10. What is the y-intercept of line k?

First, we can find the y-intercept by recalling that the y-intercept can be found by letting x = 0 and solving for y. So, we have:

⇒ 5(0) + (2/3)y = 10

⇒ 0 + 2y/3 = 10

Don’t Try to Predict Which SAT Math Topics You’ll See 

If you have been studying for the SAT for some time, my bet is that you have a solid sense of the kinds of questions you might see on the SAT. However, the exact makeup of your SAT is difficult to predict. The fact is, what you see is going to be somewhat random.

Despite College Board’s releasing numerous official SATs, you can’t assume that what you see on those exams will be exactly what you see on test day. For example, if you see a unit circle question on a practice exam, there is no guarantee that you will see one on your exam. There are 8 official practice exams that the College Board has released, for a total of 58 x 8 = 464 math questions on those exams. Your SAT will contain just 58 questions. So, there are many questions that you will have seen on practice tests that you will not see on your SAT. Likewise, those 464 practice questions do not encompass every question type that you could encounter on the SAT.

Don’t get me wrong; it’s very useful to carefully review those 8 practice exams. But if you bank your SAT math score just on those 464 questions, you will most likely have an unpleasant surprise when your SAT score is posted.

So, if you’re wondering, “What math should I practice for SAT test day?,” unfortunately, there is no specific answer to that question. What does all this mean? Well, let’s not play roulette with what you learn for the SAT. To set yourself up for success, learn all SAT math topics.

Don’t try to predict the exact makeup of your SAT.

With that point in mind, let’s discuss a great way to learn SAT math.

Thus far, we’ve discussed how learning a wide range of concepts is necessary to succeed in SAT math. A great way to learn all of these topics is through topical learning. Topical learning entails learning one topic at a time and focusing solely on that topic until you have mastered it. By studying in this way, you can ensure that you truly learn each topic before moving to the next one.

Ask yourself, would it be effective to jump from Linear Equation questions to Functions to Geometry questions? I think you know the answer …

Remember, SAT math prep takes time, care, and attention. So, jumping around from topic to topic will hinder your ability to learn.

Learn each SAT math topic one at a time.

To get a better idea of how topical learning works, let’s take a look at the Target Test Prep (TTP) study plan.

The cornerstone of the TTP study plan is topical learning and practice. The study plan is broken up into missions, each of which contains one major math topic. Students learn that topic, and then answer practice questions about that topic until they master it.

For example, mission three is the Quadratic Equations chapter. So, the first task in that mission is to learn all about quadratic equations. Those topics include FOILing quadratics, factoring quadratics, quadratic identities, the quadratic formula, completing the square, etc.

After finishing a particular section, you answer a few example questions to practice what you have just learned. Then, at the end of the chapter, you take chapter tests rated by level of difficulty, to drill every concept that was presented in that chapter.

Now that you see topical learning in action in the TTP prep course, you should have a good idea of how to structure your math studying.

In this article, we provided an overview of the SAT and presented you with 14 SAT math practice questions.

Of the four sections on the SAT, two test you on math. Section 3 is the no-calculator section, for which you must answer 20 questions in 25 minutes. Section 4 is the calculator section, and you are presented 38 questions, which you must complete in 55 minutes.

The College Board has created 4 major math conceptual topical areas: Heart of Algebra, Problem Solving and Data Analysis, Passport to Advanced Math, and Additional Topics in Math. These encompass 22 main topics, including lines, systems of equations, arithmetic, statistics, exponents, functions, and trigonometry.

To test the 22 main topics, the SAT employs two question types:

• Multiple-choice questions present 4 answer choices. Of the 58 math questions on the SAT, 45 of them are multiple-choice, including 15 in the no-calculator section and 30 in the calculator section.
• Grid-In questions require you to bubble in a 4-column answer to a question. Of the 58 math questions on the SAT, 13 of them are grid-in, including 5 in the no-calculator section and 8 in the calculator section.

There is no way to accurately predict which topics will be tested on your particular test, so you need to give attention to all of the topics. A particularly effective technique is topical studying, in which you study one topic at a time. After a topic is mastered, you move to the next one. This technique ensures that nothing slips through the cracks. You’ll be fully prepared to do your best on exam day!

Frequently Asked Questions (FAQ)

Earlier in this article, we outlined the 22 main topics that are tested on the SAT. Remember, there are many subtopics as well; SAT math encompasses a huge number of concepts that might be tested.

You will encounter easy, medium, and hard questions on the SAT. A good number of questions should be relatively easy to answer, as you saw in the practice questions in this article. But to obtain a great SAT score, you will have to study each topic thoroughly, so that you are able to answer even the most challenging problems.

Many students taking the SAT have not yet studied trigonometry, so this topic is often identified as difficult. However, the level of difficulty of the trigonometry questions is generally not high, so a quick overview of basic trig may suffice, even if you have never taken a trigonometry course.

Other areas identified as difficult by students include complex numbers, functions, and factoring of quadratics. These subtopics are usually encountered in the second year of algebra, and thus may give some students more trouble than easier topics, such as linear equations and percentages.

Now that you’ve had lots of practice with the two math question types on the SAT, you might want to read some tips for improving your SAT math score .

Additionally, from the College Board, you can access free SAT math practice tests to hone your newly acquired skills.

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About The Author

Jeffrey Miller is the head SAT instructor for Target Test Prep. Jeff has more than 14 years of experience in the business of helping students with low SAT scores hurdle the seemingly impossible and achieve the scores they need to get into the top 20 colleges in the world. Jeff has cultivated many successful college graduates through his SAT instruction, and will be a pivotal resource for many more to follow.

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Year 6 Maths SATs Past Papers and Solutions

On this page you will find all the past papers and solutions for the maths SATs papers that students in the UK sit in Year 6 (age 10 and 11). I hope they are useful to anyone revising for or teaching this exam. 

Even if you are not a primary school teacher/student, these are incredibly useful for those in secondary education. They give a great insight into the level of work students have covered at primary school, and are a great source for extra questions

2003 keyboard_arrow_up Back to Top

2004 keyboard_arrow_up back to top, 2005 keyboard_arrow_up back to top, 2006 keyboard_arrow_up back to top, 2007 keyboard_arrow_up back to top, 2008 keyboard_arrow_up back to top, 2009 keyboard_arrow_up back to top, 2010 keyboard_arrow_up back to top, 2011 keyboard_arrow_up back to top, 2012 keyboard_arrow_up back to top, 2013 keyboard_arrow_up back to top, 2014 keyboard_arrow_up back to top, 2015 keyboard_arrow_up back to top, 2016 keyboard_arrow_up back to top, 2017 keyboard_arrow_up back to top, 2018 keyboard_arrow_up back to top, 2019 keyboard_arrow_up back to top, 2022 keyboard_arrow_up back to top, 2023 keyboard_arrow_up back to top.

12 Classic SAT Math Questions You Must Know How to Solve

Most tests you take in your average math class are straightforward and obvious. As long as you’ve studied the concepts, you know what to do when you encounter a problem. You know what to expect.

On the SAT, however, even great math students may not know where to begin. Even if you know all the concepts, you won’t always know which ones to apply and how. To answer the tough questions, you need a combination of experience and a willingness to try things out.

I always warn new students that the SAT Math is unlike most tests they’ve done in school. But when they ask how it’s different, I usually have to resort to vague terms like "problem-solving, creative thinking, and word questions."

So here are 12 SAT questions that illustrate frequently tested ideas and ways of thinking that go beyond standard school material. I’ve included questions across all test topics—from algebra to geometry to logic.

All of them are tough! Try them on your own. Hints are available at the bottom and answers will be available in another post.

The Questions

• If \(2^x + 2^x + 2^x + 2^x = 2^{16}\), what is the value of \(x\)?
• The lengths of two sides of a triangle are \(4\) and \(9\). What is the greatest possible integer length of the third side?
• A car traveled \(10\) miles at an average speed of \(30\) miles per hour and then traveled the next \(10\) miles at an average speed of \(50\) miles per hour. What was the average speed, in miles per hour, of the car for the \(20\) miles?
• The average (arithmetic mean) age of a group of \(15\) teachers is \(40\) years. If \(6\) additional teachers are added to the group, then the average age of the \(21\) teachers is \(44\) years. What is the average age of the \(6\) additional teachers?
• Let the function \(f\) be defined by \(f(x) = x^2 + 18\). If \(a\) is a positive number such that \(f(2a) = 3f(a)\), what is the value of \(a\)?
• A tech company bought a number of computers for its employees: \(6\) costing \(\$1600\) each, \(4\) costing \(\$1200\) each, and \(y\) costing \(\$900\) each, where \(y\) is a positive odd integer. If the median price for all the computers purchased by the company was \(\$1200,\) what is the greatest possible value of \(y\)?

• If \(0\leq a \leq b\) and \((a + b)^2 - (a-b)^2 \geq 9\), what is the least possible value of \(b\)?
• In triangle \(ABC\), the measure of angle \(A\) is equal to \(2x^\circ + 20^\circ\) and the measure of angle \(B\) is equal to \(x^\circ + 46^\circ\). If \(\sin\angle A = \cos\angle B\), what is the value of \(x\)?
• \[f(t) = 800(1,000)^{\frac{t}{27}}\] The function \(f\) above represents the current total population of sand sharks off the coast of Greenland \(t\) years after 1980. To the nearest year, how many years did (or will) it take for the population to be 10 times larger than it was in 1980?
• In order to fulfill a prescription for 200 milliliters (mL) of a medication at a 55% solution, a pharmacist must mix \(x\) mL of a 40% solution with \(y\) mL of a 60% solution. What is the value of \(x\)?
• The answer is NOT 4. You can only add exponents when the terms are being multiplied. Think about how many \(2^{x}\)s there are.
• The answer is NOT "as big as you want." Look up the Triangle Inequality Theorem.
• The answer is NOT 40. This is much harder than that. Remember that distance \(=\) rate \(\times\) time
• Think about the sums of all the ages.
• No hint for this one. If you truly understand functions, you'll get this.
• No hint for this one either.
• What is the slope of \(\overline{OC}\)?
• Know your special triangles (30-60-90 and 45-45-90).
• First, expand (FOIL) and simply. Now how do the values of \(a\) and \(b\) affect each other when they are increased or decreased?
• You'll have to use a certain trigonometric identity.
• Use the fact that \(1,000 = 10^3\)
• 200 milliliters of a 55% solution contains how many milliliters of the actual medication?

I'm not going to post answer explanations (at least for now), because I want you to figure these out on your own. What I will do is post the answers, so you can check/review your work.

• \(\frac{75}{2}\) or \(37.5\)
• \(-\frac{3}{4}\)
• \(\frac{3}{2}\) or \(1.5\)

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Do You Understand the Problem You’re Trying to Solve?

To solve tough problems at work, first ask these questions.

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Problem solving skills are invaluable in any job. But all too often, we jump to find solutions to a problem without taking time to really understand the dilemma we face, according to Thomas Wedell-Wedellsborg , an expert in innovation and the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

In this episode, you’ll learn how to reframe tough problems by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for just one root cause can be misleading.

Key episode topics include: leadership, decision making and problem solving, power and influence, business management.

HBR On Leadership curates the best case studies and conversations with the world’s top business and management experts, to help you unlock the best in those around you. New episodes every week.

• Listen to the original HBR IdeaCast episode: The Secret to Better Problem Solving (2016)
• Find more episodes of HBR IdeaCast
• Discover 100 years of Harvard Business Review articles, case studies, podcasts, and more at HBR.org .

HANNAH BATES: Welcome to HBR on Leadership , case studies and conversations with the world’s top business and management experts, hand-selected to help you unlock the best in those around you.

Problem solving skills are invaluable in any job. But even the most experienced among us can fall into the trap of solving the wrong problem.

Thomas Wedell-Wedellsborg says that all too often, we jump to find solutions to a problem – without taking time to really understand what we’re facing.

He’s an expert in innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

  In this episode, you’ll learn how to reframe tough problems, by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for one root cause can be misleading. And you’ll learn how to use experimentation and rapid prototyping as problem-solving tools.

This episode originally aired on HBR IdeaCast in December 2016. Here it is.

SARAH GREEN CARMICHAEL: Welcome to the HBR IdeaCast from Harvard Business Review. I’m Sarah Green Carmichael.

Problem solving is popular. People put it on their resumes. Managers believe they excel at it. Companies count it as a key proficiency. We solve customers’ problems.

The problem is we often solve the wrong problems. Albert Einstein and Peter Drucker alike have discussed the difficulty of effective diagnosis. There are great frameworks for getting teams to attack true problems, but they’re often hard to do daily and on the fly. That’s where our guest comes in.

Thomas Wedell-Wedellsborg is a consultant who helps companies and managers reframe their problems so they can come up with an effective solution faster. He asks the question “Are You Solving The Right Problems?” in the January-February 2017 issue of Harvard Business Review. Thomas, thank you so much for coming on the HBR IdeaCast .

THOMAS WEDELL-WEDELLSBORG: Thanks for inviting me.

SARAH GREEN CARMICHAEL: So, I thought maybe we could start by talking about the problem of talking about problem reframing. What is that exactly?

THOMAS WEDELL-WEDELLSBORG: Basically, when people face a problem, they tend to jump into solution mode to rapidly, and very often that means that they don’t really understand, necessarily, the problem they’re trying to solve. And so, reframing is really a– at heart, it’s a method that helps you avoid that by taking a second to go in and ask two questions, basically saying, first of all, wait. What is the problem we’re trying to solve? And then crucially asking, is there a different way to think about what the problem actually is?

SARAH GREEN CARMICHAEL: So, I feel like so often when this comes up in meetings, you know, someone says that, and maybe they throw out the Einstein quote about you spend an hour of problem solving, you spend 55 minutes to find the problem. And then everyone else in the room kind of gets irritated. So, maybe just give us an example of maybe how this would work in practice in a way that would not, sort of, set people’s teeth on edge, like oh, here Sarah goes again, reframing the whole problem instead of just solving it.

THOMAS WEDELL-WEDELLSBORG: I mean, you’re bringing up something that’s, I think is crucial, which is to create legitimacy for the method. So, one of the reasons why I put out the article is to give people a tool to say actually, this thing is still important, and we need to do it. But I think the really critical thing in order to make this work in a meeting is actually to learn how to do it fast, because if you have the idea that you need to spend 30 minutes in a meeting delving deeply into the problem, I mean, that’s going to be uphill for most problems. So, the critical thing here is really to try to make it a practice you can implement very, very rapidly.

There’s an example that I would suggest memorizing. This is the example that I use to explain very rapidly what it is. And it’s basically, I call it the slow elevator problem. You imagine that you are the owner of an office building, and that your tenants are complaining that the elevator’s slow.

Now, if you take that problem framing for granted, you’re going to start thinking creatively around how do we make the elevator faster. Do we install a new motor? Do we have to buy a new lift somewhere?

The thing is, though, if you ask people who actually work with facilities management, well, they’re going to have a different solution for you, which is put up a mirror next to the elevator. That’s what happens is, of course, that people go oh, I’m busy. I’m busy. I’m– oh, a mirror. Oh, that’s beautiful.

And then they forget time. What’s interesting about that example is that the idea with a mirror is actually a solution to a different problem than the one you first proposed. And so, the whole idea here is once you get good at using reframing, you can quickly identify other aspects of the problem that might be much better to try to solve than the original one you found. It’s not necessarily that the first one is wrong. It’s just that there might be better problems out there to attack that we can, means we can do things much faster, cheaper, or better.

SARAH GREEN CARMICHAEL: So, in that example, I can understand how A, it’s probably expensive to make the elevator faster, so it’s much cheaper just to put up a mirror. And B, maybe the real problem people are actually feeling, even though they’re not articulating it right, is like, I hate waiting for the elevator. But if you let them sort of fix their hair or check their teeth, they’re suddenly distracted and don’t notice.

But if you have, this is sort of a pedestrian example, but say you have a roommate or a spouse who doesn’t clean up the kitchen. Facing that problem and not having your elegant solution already there to highlight the contrast between the perceived problem and the real problem, how would you take a problem like that and attack it using this method so that you can see what some of the other options might be?

THOMAS WEDELL-WEDELLSBORG: Right. So, I mean, let’s say it’s you who have that problem. I would go in and say, first of all, what would you say the problem is? Like, if you were to describe your view of the problem, what would that be?

SARAH GREEN CARMICHAEL: I hate cleaning the kitchen, and I want someone else to clean it up.

THOMAS WEDELL-WEDELLSBORG: OK. So, my first observation, you know, that somebody else might not necessarily be your spouse. So, already there, there’s an inbuilt assumption in your question around oh, it has to be my husband who does the cleaning. So, it might actually be worth, already there to say, is that really the only problem you have? That you hate cleaning the kitchen, and you want to avoid it? Or might there be something around, as well, getting a better relationship in terms of how you solve problems in general or establishing a better way to handle small problems when dealing with your spouse?

SARAH GREEN CARMICHAEL: Or maybe, now that I’m thinking that, maybe the problem is that you just can’t find the stuff in the kitchen when you need to find it.

THOMAS WEDELL-WEDELLSBORG: Right, and so that’s an example of a reframing, that actually why is it a problem that the kitchen is not clean? Is it only because you hate the act of cleaning, or does it actually mean that it just takes you a lot longer and gets a lot messier to actually use the kitchen, which is a different problem. The way you describe this problem now, is there anything that’s missing from that description?

SARAH GREEN CARMICHAEL: That is a really good question.

THOMAS WEDELL-WEDELLSBORG: Other, basically asking other factors that we are not talking about right now, and I say those because people tend to, when given a problem, they tend to delve deeper into the detail. What often is missing is actually an element outside of the initial description of the problem that might be really relevant to what’s going on. Like, why does the kitchen get messy in the first place? Is it something about the way you use it or your cooking habits? Is it because the neighbor’s kids, kind of, use it all the time?

There might, very often, there might be issues that you’re not really thinking about when you first describe the problem that actually has a big effect on it.

SARAH GREEN CARMICHAEL: I think at this point it would be helpful to maybe get another business example, and I’m wondering if you could tell us the story of the dog adoption problem.

THOMAS WEDELL-WEDELLSBORG: Yeah. This is a big problem in the US. If you work in the shelter industry, basically because dogs are so popular, more than 3 million dogs every year enter a shelter, and currently only about half of those actually find a new home and get adopted. And so, this is a problem that has persisted. It’s been, like, a structural problem for decades in this space. In the last three years, where people found new ways to address it.

So a woman called Lori Weise who runs a rescue organization in South LA, and she actually went in and challenged the very idea of what we were trying to do. She said, no, no. The problem we’re trying to solve is not about how to get more people to adopt dogs. It is about keeping the dogs with their first family so they never enter the shelter system in the first place.

In 2013, she started what’s called a Shelter Intervention Program that basically works like this. If a family comes and wants to hand over their dog, these are called owner surrenders. It’s about 30% of all dogs that come into a shelter. All they would do is go up and ask, if you could, would you like to keep your animal? And if they said yes, they would try to fix whatever helped them fix the problem, but that made them turn over this.

And sometimes that might be that they moved into a new building. The landlord required a deposit, and they simply didn’t have the money to put down a deposit. Or the dog might need a $10 rabies shot, but they didn’t know how to get access to a vet.

And so, by instigating that program, just in the first year, she took her, basically the amount of dollars they spent per animal they helped went from something like $85 down to around $60. Just an immediate impact, and her program now is being rolled out, is being supported by the ASPCA, which is one of the big animal welfare stations, and it’s being rolled out to various other places.

And I think what really struck me with that example was this was not dependent on having the internet. This was not, oh, we needed to have everybody mobile before we could come up with this. This, conceivably, we could have done 20 years ago. Only, it only happened when somebody, like in this case Lori, went in and actually rethought what the problem they were trying to solve was in the first place.

SARAH GREEN CARMICHAEL: So, what I also think is so interesting about that example is that when you talk about it, it doesn’t sound like the kind of thing that would have been thought of through other kinds of problem solving methods. There wasn’t necessarily an After Action Review or a 5 Whys exercise or a Six Sigma type intervention. I don’t want to throw those other methods under the bus, but how can you get such powerful results with such a very simple way of thinking about something?

THOMAS WEDELL-WEDELLSBORG: That was something that struck me as well. This, in a way, reframing and the idea of the problem diagnosis is important is something we’ve known for a long, long time. And we’ve actually have built some tools to help out. If you worked with us professionally, you are familiar with, like, Six Sigma, TRIZ, and so on. You mentioned 5 Whys. A root cause analysis is another one that a lot of people are familiar with.

Those are our good tools, and they’re definitely better than nothing. But what I notice when I work with the companies applying those was those tools tend to make you dig deeper into the first understanding of the problem we have. If it’s the elevator example, people start asking, well, is that the cable strength, or is the capacity of the elevator? That they kind of get caught by the details.

That, in a way, is a bad way to work on problems because it really assumes that there’s like a, you can almost hear it, a root cause. That you have to dig down and find the one true problem, and everything else was just symptoms. That’s a bad way to think about problems because problems tend to be multicausal.

There tend to be lots of causes or levers you can potentially press to address a problem. And if you think there’s only one, if that’s the right problem, that’s actually a dangerous way. And so I think that’s why, that this is a method I’ve worked with over the last five years, trying to basically refine how to make people better at this, and the key tends to be this thing about shifting out and saying, is there a totally different way of thinking about the problem versus getting too caught up in the mechanistic details of what happens.

SARAH GREEN CARMICHAEL: What about experimentation? Because that’s another method that’s become really popular with the rise of Lean Startup and lots of other innovation methodologies. Why wouldn’t it have worked to, say, experiment with many different types of fixing the dog adoption problem, and then just pick the one that works the best?

THOMAS WEDELL-WEDELLSBORG: You could say in the dog space, that’s what’s been going on. I mean, there is, in this industry and a lot of, it’s largely volunteer driven. People have experimented, and they found different ways of trying to cope. And that has definitely made the problem better. So, I wouldn’t say that experimentation is bad, quite the contrary. Rapid prototyping, quickly putting something out into the world and learning from it, that’s a fantastic way to learn more and to move forward.

My point is, though, that I feel we’ve come to rely too much on that. There’s like, if you look at the start up space, the wisdom is now just to put something quickly into the market, and then if it doesn’t work, pivot and just do more stuff. What reframing really is, I think of it as the cognitive counterpoint to prototyping. So, this is really a way of seeing very quickly, like not just working on the solution, but also working on our understanding of the problem and trying to see is there a different way to think about that.

If you only stick with experimentation, again, you tend to sometimes stay too much in the same space trying minute variations of something instead of taking a step back and saying, wait a minute. What is this telling us about what the real issue is?

SARAH GREEN CARMICHAEL: So, to go back to something that we touched on earlier, when we were talking about the completely hypothetical example of a spouse who does not clean the kitchen–

THOMAS WEDELL-WEDELLSBORG: Completely, completely hypothetical.

SARAH GREEN CARMICHAEL: Yes. For the record, my husband is a great kitchen cleaner.

You started asking me some questions that I could see immediately were helping me rethink that problem. Is that kind of the key, just having a checklist of questions to ask yourself? How do you really start to put this into practice?

THOMAS WEDELL-WEDELLSBORG: I think there are two steps in that. The first one is just to make yourself better at the method. Yes, you should kind of work with a checklist. In the article, I kind of outlined seven practices that you can use to do this.

But importantly, I would say you have to consider that as, basically, a set of training wheels. I think there’s a big, big danger in getting caught in a checklist. This is something I work with.

My co-author Paddy Miller, it’s one of his insights. That if you start giving people a checklist for things like this, they start following it. And that’s actually a problem, because what you really want them to do is start challenging their thinking.

So the way to handle this is to get some practice using it. Do use the checklist initially, but then try to step away from it and try to see if you can organically make– it’s almost a habit of mind. When you run into a colleague in the hallway and she has a problem and you have five minutes, like, delving in and just starting asking some of those questions and using your intuition to say, wait, how is she talking about this problem? And is there a question or two I can ask her about the problem that can help her rethink it?

SARAH GREEN CARMICHAEL: Well, that is also just a very different approach, because I think in that situation, most of us can’t go 30 seconds without jumping in and offering solutions.

THOMAS WEDELL-WEDELLSBORG: Very true. The drive toward solutions is very strong. And to be clear, I mean, there’s nothing wrong with that if the solutions work. So, many problems are just solved by oh, you know, oh, here’s the way to do that. Great.

But this is really a powerful method for those problems where either it’s something we’ve been banging our heads against tons of times without making progress, or when you need to come up with a really creative solution. When you’re facing a competitor with a much bigger budget, and you know, if you solve the same problem later, you’re not going to win. So, that basic idea of taking that approach to problems can often help you move forward in a different way than just like, oh, I have a solution.

I would say there’s also, there’s some interesting psychological stuff going on, right? Where you may have tried this, but if somebody tries to serve up a solution to a problem I have, I’m often resistant towards them. Kind if like, no, no, no, no, no, no. That solution is not going to work in my world. Whereas if you get them to discuss and analyze what the problem really is, you might actually dig something up.

Let’s go back to the kitchen example. One powerful question is just to say, what’s your own part in creating this problem? It’s very often, like, people, they describe problems as if it’s something that’s inflicted upon them from the external world, and they are innocent bystanders in that.

SARAH GREEN CARMICHAEL: Right, or crazy customers with unreasonable demands.

THOMAS WEDELL-WEDELLSBORG: Exactly, right. I don’t think I’ve ever met an agency or consultancy that didn’t, like, gossip about their customers. Oh, my god, they’re horrible. That, you know, classic thing, why don’t they want to take more risk? Well, risk is bad.

It’s their business that’s on the line, not the consultancy’s, right? So, absolutely, that’s one of the things when you step into a different mindset and kind of, wait. Oh yeah, maybe I actually am part of creating this problem in a sense, as well. That tends to open some new doors for you to move forward, in a way, with stuff that you may have been struggling with for years.

SARAH GREEN CARMICHAEL: So, we’ve surfaced a couple of questions that are useful. I’m curious to know, what are some of the other questions that you find yourself asking in these situations, given that you have made this sort of mental habit that you do? What are the questions that people seem to find really useful?

THOMAS WEDELL-WEDELLSBORG: One easy one is just to ask if there are any positive exceptions to the problem. So, was there day where your kitchen was actually spotlessly clean? And then asking, what was different about that day? Like, what happened there that didn’t happen the other days? That can very often point people towards a factor that they hadn’t considered previously.

SARAH GREEN CARMICHAEL: We got take-out.

THOMAS WEDELL-WEDELLSBORG: S,o that is your solution. Take-out from [INAUDIBLE]. That might have other problems.

Another good question, and this is a little bit more high level. It’s actually more making an observation about labeling how that person thinks about the problem. And what I mean with that is, we have problem categories in our head. So, if I say, let’s say that you describe a problem to me and say, well, we have a really great product and are, it’s much better than our previous product, but people aren’t buying it. I think we need to put more marketing dollars into this.

Now you can go in and say, that’s interesting. This sounds like you’re thinking of this as a communications problem. Is there a different way of thinking about that? Because you can almost tell how, when the second you say communications, there are some ideas about how do you solve a communications problem. Typically with more communication.

And what you might do is go in and suggest, well, have you considered that it might be, say, an incentive problem? Are there incentives on behalf of the purchasing manager at your clients that are obstructing you? Might there be incentive issues with your own sales force that makes them want to sell the old product instead of the new one?

So literally, just identifying what type of problem does this person think about, and is there different potential way of thinking about it? Might it be an emotional problem, a timing problem, an expectations management problem? Thinking about what label of what type of problem that person is kind of thinking as it of.

SARAH GREEN CARMICHAEL: That’s really interesting, too, because I think so many of us get requests for advice that we’re really not qualified to give. So, maybe the next time that happens, instead of muddying my way through, I will just ask some of those questions that we talked about instead.

THOMAS WEDELL-WEDELLSBORG: That sounds like a good idea.

SARAH GREEN CARMICHAEL: So, Thomas, this has really helped me reframe the way I think about a couple of problems in my own life, and I’m just wondering. I know you do this professionally, but is there a problem in your life that thinking this way has helped you solve?

THOMAS WEDELL-WEDELLSBORG: I’ve, of course, I’ve been swallowing my own medicine on this, too, and I think I have, well, maybe two different examples, and in one case somebody else did the reframing for me. But in one case, when I was younger, I often kind of struggled a little bit. I mean, this is my teenage years, kind of hanging out with my parents. I thought they were pretty annoying people. That’s not really fair, because they’re quite wonderful, but that’s what life is when you’re a teenager.

And one of the things that struck me, suddenly, and this was kind of the positive exception was, there was actually an evening where we really had a good time, and there wasn’t a conflict. And the core thing was, I wasn’t just seeing them in their old house where I grew up. It was, actually, we were at a restaurant. And it suddenly struck me that so much of the sometimes, kind of, a little bit, you love them but they’re annoying kind of dynamic, is tied to the place, is tied to the setting you are in.

And of course, if– you know, I live abroad now, if I visit my parents and I stay in my old bedroom, you know, my mother comes in and wants to wake me up in the morning. Stuff like that, right? And it just struck me so, so clearly that it’s– when I change this setting, if I go out and have dinner with them at a different place, that the dynamic, just that dynamic disappears.

SARAH GREEN CARMICHAEL: Well, Thomas, this has been really, really helpful. Thank you for talking with me today.

THOMAS WEDELL-WEDELLSBORG: Thank you, Sarah.  

HANNAH BATES: That was Thomas Wedell-Wedellsborg in conversation with Sarah Green Carmichael on the HBR IdeaCast. He’s an expert in problem solving and innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

We’ll be back next Wednesday with another hand-picked conversation about leadership from the Harvard Business Review. If you found this episode helpful, share it with your friends and colleagues, and follow our show on Apple Podcasts, Spotify, or wherever you get your podcasts. While you’re there, be sure to leave us a review.

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This episode was produced by Anne Saini, and me, Hannah Bates. Ian Fox is our editor. Music by Coma Media. Special thanks to Maureen Hoch, Adi Ignatius, Karen Player, Ramsey Khabbaz, Nicole Smith, Anne Bartholomew, and you – our listener.

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