Learn›Arithmetic ReasoningAR›Averages
Averages and remainders
Finding an average is easy. The test asks the harder questions: what number is missing, what score you need next, how two groups average together, and what to do with a remainder. They all come back to one idea: the total.
An average is the level you get when you even things out
average = sum ÷ count
Add up all the values and share the total equally among them. That equal share is the average (also called the mean). The total is what really matters: almost every hard average question is solved by finding it.
Turn the average back into a total
If you know the average and how many values there are, you know the total: sum = average × count. That one move solves missing numbers, the score you need and groups of different sizes.
How to solve them
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1
Average = sum ÷ count
Add every value, then divide by how many values there are. A quick check: the average always lands between the smallest and the largest value.
Next step: The hidden sumExampleTest scores of 78, 85, 91, 70 and 86. What is the average?
- Sum: 78 + 85 + 91 + 70 + 86 = 410.
- Count: 5 scores.
- 410 ÷ 5 = 82, between 70 and 91 as it should be.
Answer 82
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2
Work backwards: sum = average × count
When a value is missing, find the total the average demands, then take away the values you know. What is left is the missing one.
Next step: The score you needExampleThe average of 6 numbers is 14. Five of them add up to 66. What is the sixth?
- Total of all six: 14 × 6 = 84.
- The five known add to 66.
- 84 − 66 = 18.
Answer 18
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3
The score you need: target total minus what you have
Multiply the average you want by the total number of tests. That is the total you need. Subtract the points you already have.
Next step: Weighted averageExampleKara scored 82, 88 and 79 on three tests. What does she need on the fourth to average 85?
- Needed total: 85 × 4 = 340.
- She has 82 + 88 + 79 = 249.
- 340 − 249 = 91.
Answer 91
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4
Groups of different sizes: weigh each average
Two groups of different sizes do not average halfway between their averages. The bigger group pulls the result toward its own average. Turn each group into a total, add the totals, and divide by everyone.
Next step: RemaindersExample12 students averaged 80 and 8 students averaged 90. What is the average of all 20?
- Totals: 12 × 80 = 960 and 8 × 90 = 720.
- All together: 960 + 720 = 1,680 points for 20 students.
- 1,680 ÷ 20 = 84, not 85: the bigger group pulls it down.
Answer 84
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5
Remainders: round up or down, depending on the question
A division with a leftover can have three right answers, and the question decides which. If everyone needs a place, round up. If you can only buy whole things, round down. Sometimes the question wants the leftover itself.
Next step: Same unitsRound up58 people, 8 per car. How many cars?
58 ÷ 8 = 7 r 2 → 8 cars
Round down$58, $8 per ticket. How many tickets?
7 r 2 → 7 tickets
The leftover58 cookies in bags of 8. How many are left over?
2 cookies
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6
Comparing amounts: same units first
Before you compare, add or average two amounts, put them in the same unit. These come up all the time:
1 foot=12 inches1 yard=3 feet1 hour=60 minutes1 pound=16 ounces1 gallon=4 quartsExampleRope A is 2 yards long and rope B is 70 inches. Which is longer, and by how much?
- 2 yards = 2 × 3 = 6 feet = 6 × 12 = 72 inches.
- 72 − 70 = 2.
Answer Rope A, by 2 inches
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Averaging two averages
30 workers earn $15 an hour and 10 earn $25. The average for all 40:
Wrong(15 + 25) ÷ 2 = $20
Right(450 + 250) ÷ 40 = $17.50
Rounding the wrong way
45 campers, 6 per tent. How many tents?
Wrong7 tents
Right45 ÷ 6 = 7.5 → 8 tents
Forgetting the new count
Four scores average 80. A fifth score of 90 is added:
Wrong(80 + 90) ÷ 2 = 85
Right(320 + 90) ÷ 5 = 82
Mixing units
The average of 1 hour and 30 minutes:
Wrong(1 + 30) ÷ 2 = 15.5
Right(60 + 30) ÷ 2 = 45 minutes
5 questions at test level
Pick an answer, then read why. No timer, no account.
The average of 7 numbers is 18. What is their sum?
Show the answer
C. Sum = average × count = 18 × 7 = 126. 25 adds the two numbers, 2.6 divides them, and 144 multiplies by 8. The sum is 126.
Jamal’s first four quiz scores are 76, 84, 90 and 82. What must he score on the fifth quiz to average 85?
Show the answer
D. To average 85 over 5 quizzes he needs 85 × 5 = 425 points. He has 76 + 84 + 90 + 82 = 332. 425 − 332 = 93. 83 is his current average. He needs 93.
A van holds 9 passengers. How many vans are needed to take 75 people on a trip?
Show the answer
D. 75 ÷ 9 = 8 with 3 left over. Those 3 people still need seats, so round up. 8 vans would leave 3 people behind, and 3 is only the leftover. They need 9 vans.
At a company, 15 employees average 40 hours a week and 5 employees average 24 hours a week. What is the average for all 20 employees?
Show the answer
B. Weigh each group by its size: 15 × 40 = 600 hours and 5 × 24 = 120 hours, so 720 hours for 20 people, and 720 ÷ 20 = 36. 32 averages the two averages as if the groups were the same size. The average is 36.
A board 7 feet 3 inches long is cut into 3 equal pieces. How long is each piece?
Show the answer
B. 7 feet 3 inches = 7 × 12 + 3 = 87 inches, and 87 ÷ 3 = 29. 24 inches drops the 3 extra inches, and 87 inches is the whole board. Each piece is 29 inches.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
You finished averages and remainders.
Practice score: not taken yet Take the practice
Next topic: SequencesSee this topic move.
In the course, every topic is drawn step by step on a whiteboard while a voice explains it, with timed practice tests and a plan for your test date. The first lessons of every module are free.
