Hard 5th grade math questions: 35 Math Questions For 5th Graders: Worked Examples

35 Math Questions For 5th Graders: Worked Examples

In 5th grade math, the toughest questions are often the reasoning questions. In this article, we’ve put together a collection of math questions for 5th graders, organized by the different kinds of reasoning questions that students may encounter on standardized tests and beyond.

End of Year Math Assessments Grades 4 and 5

Test your students with this free pack of 4th and 5th grade end of year assessments

Why Focus On Math Reasoning Questions?

Most fifth graders find reasoning questions to be the most difficult. Unsurprisingly, we teach thousands of students in the weeks leading up to standardized tests. Teaching them math reasoning skills at the elementary level is a big part of what we do here at Third Space Learning.

We even recently made the decision to restructure our elementary lessons to introduce math reasoning questions earlier in their learning journey as the difficulty level was just too high at the end of the lesson. We  definitely feel fifth grade teachers’ pain!

Whatever level your students are currently achieving in math, math reasoning questions will appear from elementary to high school, so it is an essential skill for the future.

If you find you have children in your class with a lot more catching up to do than others, then we may be able to support them with some personalized one-on-one tutoring if you get in touch.

35 Math Questions For 5th Graders

There are 7 types of math reasoning questions that fifth graders are likely to encounter:

For each of these types we’ll examine an example problem, looking at the question, the correct answer, and how to go about answering this problem.

We’ll also look at further examples of each type of math reasoning question and answer, again with worked examples and an explanation of how to answer each.

Our aim is to provide you with a sample of the types of math reasoning questions and how to teach the reasoning and problem solving skills they’ll need to solve them.

For more word problems like this, check out our collection of 2-step and multi-step word problems. For advice on how to teach children to solve problems like this, check out these math problem solving strategies.

Math Question Type 1: Single step word problems

The simplest type of reasoning question students are likely to encounter, single step problems are exactly that: students are asked to interpret a written question and carry out a single mathematical step to solve it.

Take a look at the question below:

Reasoning Question 1

Answer: $0.65

A relatively easy question to interpret–the first step will be to rewrite the amounts given so that they can properly line up the place values in order to solve. From here the simple mathematical step is subtraction i.e. $2.00 – $1.35 = 0.65.

The most crucial skill for grade schoolers in this question is a solid understanding of money as relating to place value. If this understanding is present, the mathematical step itself is quite easy.

Below are several more examples:

Reasoning Question 2

Answer: 7 hours 24 minutes

Students need to understand that one hour is equal to 60 minutes. From here the single mathematical step is division: 444/60, to find a whole number answer with a remainder.

Reasoning Question 3

Answer: 48 cm³

Students must multiply length by width by height, using the amounts provided by the question.

Reasoning Question 4

Answer: 1,488 cubic cm

A simple enough calculation (multiplying) if students are aware that the volume of a rectangular prism can be found by multiplying the area of the base by the height.

Reasoning Question 5

Answer: 7,590

A single, relatively simple rounding problem – students should recognize that ’94’ is the place they should focus on for this problem.

Math Question Type 2: Multiple step worded problems

A more complex version of the single step word problem, multi-step problems require students to interpret a written problem, but solving it then requires the use of two or three math skills.

For example, consider this question below:

Reasoning Question 1

Answer: $1.85

This question encompasses three different math skills: multiplying (and dividing) decimals, addition and subtraction. Students can choose to work out the multiplication or division first, but must complete both before moving on.

Once these values have been worked out the next steps are relatively simple – adding the two values together, and subtracting the total from $5.

Multi-step problems are particularly valuable to include in practice tests because they require children to apply their knowledge of math language and their reasoning skills several times across the course of a single question, usually in slightly different contexts.

More examples:

Reasoning Question 2

Answer: $5,520

There are two steps to this problem, but both are multiplication. The first is to work out how much money is made per day – 92 x $15. This product is then multiplied by 4 – the number of days – to get to the answer.

Reasoning Question 3

Answer: 1360 miles

Another two step problem. The first step is to work out 4 of 3,400 miles. Then divide this by 10 to solve for 4/10 of 3,400.

Reasoning Question 4

Answer: $153

There are four steps involved in solving this problem: multiplication (doubling $51), division (dividing $51 in half), multiplication again (doubling half of $51–which some students may recognize those last two steps were unnecessary as that brings us back to $51), and addition (putting the two costs together).

Given the number of steps involved it can be easy for students to make arithmetic mistakes.

Reasoning Question 5

Answer: 11.45 lbs

A two-step problem again: multiplying 3.45 lbs by 4, then subtracting 2.35 lbs from the total. 

Math Question Type 3: Problems involving measurements

As their name suggests, these questions ask students to solve a problem that includes one or more units of measurement.

A slide from a Third Space Learning 1-to-1 tutoring lesson which teaches reading units of measure.

Reasoning Question 1

Answer: 40 washes

This is a two step problem; students must first be able to read and convert kilograms to grams (and therefore know the relationship and conversions between the two units- 1,000 grams to 1 kilogram), multiply 2.6 by 1,000 which equals 2,600, then divide 2,600 by 65. The quotient is the number of washes possible.

Further examples:

Reasoning Question 2

Answer: 50g

A relatively simple division problem, relying on students having knowledge that 200g is one fifth of a kilogram.

Reasoning Question 3

Answer: 5.12 miles

Another three step problem, and this requires students to subtract and divide decimals – subtracting 12.63 miles from the total amount, taking the difference, 13.91, and subtracting 3.67 miles, and then dividing that difference, 10.24, in half to obtain the distance the other two friends ran.

Reasoning Question 4

Answer: 84 inches/7 foot

To find 8 feet in inches, students must multiply 8 by 12. This gives the answer 96 inches. Students must then divide 96 by 40 to find the height of one box: 2.4 inches. Multiply 2.4 by 5 and minus this from the original 96 inch tower. 

Interesting to note that the units for the answer may or may not be specified – an answer given in inches or feet will be accepted, however sometimes the unit will be specified in the answer box. This is why we encourage students to keep an eye on whether units are provided in the answer box.

Reasoning Question 5

Answer: 0.05 lbs

As with the running question there are three steps involved to solve this problem: subtracting the heaviest car from the total amount (3.85 – 1), figuring out the weight of the remaining three cars (2.85/3) and subtracting 0.95 from 1 to get the remaining amount of 0.05 lbs.

Question Type 4: Problems involving drawing

Problems involving drawing require students to construct an accurate drawing by following a set of instructions, or through reflection, translation, or scaling.  

Reasoning Question 1

Answer: Any pair of lines that make a square of 4 units, a rectangle of 6 units, and a square of 25 units.

This question is considerably more complex than it appears, and incorporates aspects of multiplication as well as spatial awareness. One potential solution is to work out the area of the card (35), then work out the possible square numbers that will fit in (understanding that square numbers produce a square when drawn out as on a grid), and which then leave a single rectangle behind.

A lot of work for a single point!

Some further examples:

Reasoning Question 2

Answer: Any quadrilateral made by joining the dots that has 3 acute angles e.g. an arrowhead shape.

Reasoning Question 3

Answer: An accurately drawn angle.

This question demands students to have an understanding of and ability to accurately use a protractor. Often, a mark scheme allows some room for error – “between 34 and 36 degrees” is acceptable.

Reasoning Question 4

Answer: An accurately drawn angle.

As with the question above, a small amount of room for error is given as it acceptable to be between 139 and 141 degrees.

Reasoning Question 5

Answer: Points drawn at (2,1), (5,1) and (2,4).

Math Question Type 5: Explanation questions

These problems ask children to explain a mathematical statement or error.

As an example:

Reasoning Question 1

Answer: If the distance from P to R is 800 yards and the distance from P to Q is (Q -> R x 4), it must be 4/5 of 800 = 640 yards. Therefore Olivia is wrong.

More than most problems, this type requires students to actively demonstrate their reasoning skills as well as their mathematical ones. Here students must articulate either in words or (where possible) numerically that they understand that Q to R is 1/5 of the total, that therefore P to Q is 4/5 of the total distance, and then calculate what this is via division and multiplication.

Further examples below:

Reasoning Question 2

Answer: No; 20/100 is the same as 20 divided by 100, which equals 0.2

Reasoning Question 3

Answer: No; multiplication and division have the same priority in the order of operations, so in a problem like 40 x 6 ÷2, you would carry out the multiplication first as it occurs first.

Reasoning Question 4

Answer: No

Any explanation that provides a counter-example is acceptable e.g. “Not if the number is 1”, “Not for 0,” “Not if the number is less than 1” etc.

Reasoning Question 5

Answer: Any answer that refers to the fact that there is a 5 in the hundredths place, AND a 9 in the thousandths place, so that the number has to be rounded up as far as the ten-thousands place.

Math Question Type 6: Sequence questions

Another relatively simple kind of reasoning question, sequence problems involve students completing mathematical sequences.

Consider this example:

Reasoning Question 1

Answer: 35, 42, 49, 56, 63, 70

The question’s instructions point clearly to the solution: figure out what the increase between numbers is, then apply this via addition or subtraction to find the missing numbers.

Higher achieving students might quickly pick up that this is in fact the 7 times table and rely on their knowledge of multiplication facts to obtain the answer – this should be encouraged so long as they then check their answer in the normal method to ensure they haven’t made a mistake.

More examples:

Reasoning Question 2

Answer(s): 5/8 and 2 1/8 (OR 17/8)

Both answers must be correct to receive the point. Students must recognize that 3/4 is the same as 6/8, so that the following number must be three eighths higher. They then must be able to add and subtract fractions to obtain the answers.

Reasoning Question 3

Answer(s): 4.2 and 7

Reasoning Question 4

Answer(s): 128, 135 and 156.

Reasoning Question 5

Answer(s): 0 and 24

This number line question can be a little tricky; students need to figure out that the marks on the line represent increments of 1½, and count backwards and forwards in 1½’s to obtain the missing numbers.

Math Question Type 7: Ordering questions

A slightly more complex variation of the sequence question, ordering problems require students to put a set of numbers, fractions or measures in the correct order.

A good example is this fifth grade math question below:

Reasoning Question 1

Answer: 3/5, 3/4, 6/5

This question throws a wrench in things by including an improper fraction, but this is hardly unusual. These sorts of questions are just the place to find other ‘curveballs’ such as equivalent fractions, mixed numbers, decimal numbers, and fractions all mixed into one problem.

A good knowledge of the fundamentals of fractions is essential here: students must understand what a larger denominator means, and the significance of a fraction with a numerator greater than its denominator.

Further examples:

Reasoning Question 2

Answer: D,C,A,B

Encourage students to convert all the fractions to one denominator value to make ordering easier.

Reasoning Question 3

Answer: (descending down the ‘Place’ column) 3rd, 5th, 2nd, 4th

Students could use many strategies to solve this problem. The most time consuming would be to rewrite all the fractions with a common denominator. More efficient strategies would include reasoning about the size of the fractions in comparison to ½ or 1. For instance, a student may notice that ⅜ is the only fraction less than ½, putting Ben in 5th place. 4/8 is exactly ½ whereas the others are greater than ½, putting Michael in 4th place. Then the student may recognize that 10/12 is closer to 1 than ¾, completing the rest of the table.

Reasoning Question 4

Answer: C, B, D, A

Reasoning Question 5

Answer: D, A, C, B

7 Top Tips For Answering 5th Grade Math Reasoning Questions

Now that we’ve covered how to answer some specific types of reasoning questions, here are some more generic tips for success in standardized tests. They may not all be applicable to every single question, but will apply to at least two, usually more.

• Get students in the habit of identifying what information they’re given in a question, and what they need to know to solve the problem. This helps them start to form the steps needed to find the solution.
• Ask students to ‘spot the math’ in a question – which operations or skills do they actually need to use to solve the problem? This is useful even for arithmetic questions – it’s no surprise how often children can misread a question.
• Check the units! Especially in questions involving multiple measures, it can be easy to give the answer in the wrong one. The answer box might give a specific unit of measurement, so students should work to give their answer in that unit.
• In a similar vein, remind students to convert different units of measurement in a question into the same unit to make calculations easier e.g. lbs to oz.
• Encourage numerical answers where possible. Even in explanation questions demonstrating the mathematical equation is a better explanation than trying to write it out.
• The bar model can be a useful way of visualizing many different types of questions, and might make it easier to spot the ‘steps’ needed for the solution.
• Check your work! Even if the work is ultimately irrelevant to the question, you can lose points if it is wrong.

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The content in this article was originally written by Anantha Anilkumar at Third Space Learning and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

5 Grade School Math Problems That Are So Hard, You’ll Wonder How You Ever Made it To High School

A math problem can often look super simple… before you sit down to actually do it and find you have no clue how to solve it. Then there are the problems that make you feel like a math whiz when you solve it in 2 seconds flat — only to find your answer is WAAAAY off. That’s why math problems go viral all the time, because they’re simultaneously easy and yet so not.

Here are five problems that prove the point:

1. What’s the Question Mark?

Let’s start off super simple. Can you solve what number the question mark is supposed to be?

The Answer: 6.

Explanation: All of the rows and columns should add up to 15.

2. The Bat & The Ball

A bat and a ball cost one dollar and ten cents in total. The bat costs a dollar more than the ball. How much does the ball cost?

Was your answer 10 cents? That would be wrong!

The Answer: The ball costs 5 cents.

Explanation: When you read the math problem, you probably saw that the bat and the ball cost a dollar and ten cents in total and when you processed the new information that the bat is a dollar more than the ball, your brain jumped to the conclusion that the ball was ten cents without actually doing the math. But the mistake there is that when you actually do the math, the difference between $1 and 10 cents is 90 cents, not $1. If you take a moment to actually do the math, the only way for the bat to be a dollar more than the ball AND the total cost to equal $1.10 is for the baseball bat to cost $1.05 and the ball to cost 5 cents.

3. To Switch or Not to Switch

Imagine you’re on a game show, and you’re given the choice of three doors: Behind one door is a million dollars, and behind the other two, nothing. You pick door #1, and the host, who knows what’s behind the doors, opens another door, say #3, and it has nothing behind it. He then says to you, «Do you want to stick with your choice or switch?»

So, is it to your best advantage to stick with your original choice or switch your choice?

Most people think the choice doesn’t matter because you have a 50/50 chance of getting the prize whether you switch or not since there are two doors left, but that’s actually not true!

The Answer: You should always switch your choice!

The Explanation: When you first picked one of the three doors, you had a 1 in 3 chance of picking the door with the prize behind it, which means you had a 2 in 3 chance of picking an empty door. What people get wrong here is thinking that because there are only two doors left in play, you have a 50% chance your first choice was correct. In actuality, your chances never changed.

There’s still a 1 in 3 chance you picked the right door and a 2 in 3 chance you picked an empty door, which means that when the host opened one of the empty doors, he eliminated one of the WRONG choices and the chances that the prize is behind the last closed door is still 2 in 3 — double what the chances you picked the right door at first are. So, basically, by switching your door choice, you’re betting on the 2 in 3 chance you picked the wrong door at first.

Sure, you aren’t guaranteed to win if you switch, but if you play the game over and over, you’ll win 2/3rds of the time using this method!

Still confused? Let the genius UC Berkeley math professor Lisa Goldberg explain it even better with a bunch of diagrams!

View full post on Youtube

4.

The PEMDAS Problem

When you do this seemingly simple problem, what is the answer you get?

The masses are split on the answer to this stumper. Some people are POSITIVE the answer is 1 and some people are absolutely sure the answer is 9.

The Answer: The winner is — 9!

Explanation: The handy order of operations rule you learned in grade school, PEMDAS, says you should solve a problem by working through the Parentheses, then the Exponents, the Multiplication and Division, followed by Addition and Subtraction. But the thing about PEMDAS is, some people interpret it different ways and in there lies the controversy behind this problem.

Some people think that anything touching a parentheses should be solved FIRST. Which means they simplify the problem as follows: 6÷2(1+2) = 6÷ 2(3) = 6÷6 = 1.

But just because a number is touching a parentheses doesn’t mean it should be multiplied before division that’s to the left of it. PEMDAS says to solve anything inside parentheses, then exponents, and then all multiplication and division from left to right in the order both operations appear (that’s the key). That means that once you solve everything inside the parenthesis and simplify the exponents, you go from left to right no matter what. That means the problem should actually be solved as follows: 6÷2(1+2) = 6÷2*(1+2) = 6÷2*3 = 3*3 = 9.

5. The Lily Pad Problem

In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

The tempting answer here is 24, but you’re wrong if that’s your final answer!

The Answer: The patch would reach half the size of the lake on day 47.

Explanation: With all the talk of doubling and halves, your brain jumps to the conclusion that to solve the problem of when the lily patch covers half the lake, all you have to do is divide the number of days it took to fill the lake (48) in half. It’s understandable but wrong.

The problem says that the patch DOUBLES in size every day, which means that on any day, the lily patch was half the size the day before. So if the patch reaches the entire size of the lake on the 48th day, it means the lily pad was half the size of the lake on day 47.

Noelle Devoe

Entertainment Editor

When I’m not holed up in my room going on a completely unproductive Netflix binge or Tumblr stalking Timothée Chalomet, I’m searching for awesome celeb news stories that Seventeen readers will love!

Mathematics tests for grade 5 on the topic «Percentage» online

• Calculating percentages of a number (50, 100, 150, 200, 300)
  

  09/19/2020
  1918
  0

  Percentage workout for a given number (50, 100, 150, 200 and 300). There are 11 tasks in the test, which are randomly selected from the total base — 100 tasks. Rating «5» — for 91-100%, «4» — for 70-90%, «3» — for 50-69% correct answers.

• Interest. Problem solving
  

  04/21/2020
  5260
  0

  The test on the topic «Percentage. Solving problems» is intended for students in grades 5-6, contains text tasks on the topic «Percentage calculations»

• Test on the topic «Percentage»
  

  03/31/2020
  6094

  The test was created to generalize and systematize knowledge on the topic «Interest».

• Finding a number by percentage Grade 5
  

  04/14/2020
  8383
  0

  This test is designed to consolidate the material on the topic «Interest». Read the assignment and instructions very carefully. Good luck!!!

• Interest, interest calculation
  

  06/09/2021
  854
  0

  Test for mastering the initial concept on the topic «Proceets», for solving the simplest problems of converting numbers into percentages and percents into fractions, solving simple problems for calculating percentages

• Finding the percentage of a number Grade 5
  

  04/12/2020
  13023

  This test is designed to consolidate the material on the topic «Interest». Read the assignment and instructions very carefully. Good luck!!!

• Word problems for percentages (with decimals)
  

  09/19/2020
  1557
  0

  Tasks for fixing the material on the topic «Interest». In the test, 5 tasks are randomly selected from the general database of tasks on the topic. For each correctly completed task, 1 point is awarded. At the end of the test, the result and grade are immediately visible. Criteria: «3» — 3 points, «4» — 4 points, «5» — 5 points.

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This catalog contains interactive computer tests in «Mathematics» for grade 5. Any test that is on our portal can be downloaded and used on your local computer, or you can solve and check answers directly on the site

Test: Signs of Divisibility No. 2

Test on the topic «Signs of Divisibility» for 5th grade students.

Mathematics Grade 5 | Author: Saravas E.F. | ID: 17724 | Date: 6.3.2023

Test: Quiz «Tsifrovest»

Test quiz

Mathematics Grade 5 | Author: Afanasyeva Irina | ID: 17412 | Date: 6.9.2022

Test: Multiplication of decimal fractions №3

complete the test by choosing 1 correct answer

Mathematics Grade 5 | Author: Bochkareva Tatiana | ID: 17411 | Date: 09/06/2022

Test: Conversion

Take the test by choosing 1 answer

Mathematics Grade 5 | Author: Bochkareva Tatiana | ID: 17410 | Date: 6.9.2022

Test: Rounding decimals №3

complete the test by choosing 1 correct answer

Mathematics Grade 5 | Author: Bochkareva Tatiana | ID: 17409 | Date: 6.