Get the app
Learn›FoundationsBasics›Mental math
Foundations topics All Foundations topics

Mental math and estimating

Pythagoras holding a basket of apples at a sunny market, counting in his head

The ASVAB gives you no calculator, and the clock keeps running. You do not need to be a human calculator. You need a few tricks that turn hard arithmetic into easy arithmetic, and the habit of estimating first, so the answer choices do half the work for you.

13 min read · 15 practice questions · 12 flashcards · Free

Mental math means changing the problem, not the answer

  1. 38 + 47
  2. 40 + 45
  3. 85
Move 2 from the 47 to the 38. The total does not change, but now the problem is easy.
8 is 2 squares away from a full ten. That is why 8 + 7 is easier as 8 + 2 + 5: first make the ten, then add what is left. In 38 + 47, the 2 comes from the 47.

Mental math is doing arithmetic in your head, without paper or a calculator. The secret is not to be fast at hard problems. It is to turn a hard problem into an easy one that has the same answer.

Key idea

Estimate first

An estimate is a rough answer. You get it by rounding: changing each number to a close, easy one, like 49 to 50. On a multiple-choice test, an estimate often shows you the answer before you calculate, because some of the choices are far from it.

Next: Learn

How to solve them

  1. 1

    Adding and subtracting: make a ten

    Numbers that end in 0 are easy to add and subtract. So before you start, look for a number that is close to a ten (a number like 30, 40 or 60) and move a little to reach it.

    In addition, you can move an amount from one number to the other: the total stays the same. In subtraction it works another way: you add the same amount to both numbers, and the difference (the gap between them) stays the same.

    1. 57 + 36
    2. 57 + 3 + 36 − 3
    3. 60 + 33
    4. 93
    Addition: give 3 from the 36 to the 57.
    1. 83 − 37
    2. 86 − 40
    3. 46
    Subtraction: add 3 to both numbers. The gap between them does not change.
    Next step: Split it
  2. 2

    Multiplying: split a number into easy parts

    To multiply by a two-digit number, split it into tens and ones, multiply each part, and add. This works because 7 × 46 means seven 46s, and each 46 is a 40 and a 6.

    When a number is just below a round one, like 98 or 199, it is easier to go up to the round number and then take back the extra.

    1. 7 × 46
    2. 7 × (40 + 6)
    3. 7 × 40 + 7 × 6
    4. 280 + 42
    5. 322
    1. 6 × 98
    2. 6 × (100 − 2)
    3. 600 − 12
    4. 588
    Go up to 100, then take back 6 × 2.
    Next step: ×5 and ×25
  3. 3

    ×5, ×25 and ×50: use 10 and 100

    5 is half of 10. So multiplying by 5 is the same as multiplying by 10 and taking half. In the same way, 25 is a quarter of 100, and 50 is half of 100.

    × 5

    Times 10, then take half.

    68 × 10 = 680; half is 340
    × 25

    Times 100, then take a quarter.

    44 × 100 = 4,400; a quarter is 1,100
    × 50

    Times 100, then take half.

    26 × 100 = 2,600; half is 1,300
    1. 68 × 5
    2. 68 × 10 ÷ 2
    3. 680 ÷ 2
    4. 340
    1. 44 × 25
    2. 44 × 100 ÷ 4
    3. 4,400 ÷ 4
    4. 1,100
    To take a quarter, halve twice: 4,400, then 2,200, then 1,100.
    Shortcut

    Halve first

    If the other number is even, take half of it first. 68 × 5 is 34 × 10, which is 340.

    Next step: ×9 and ×11
  4. 4

    ×9 and ×11: ten times, then one more or one less

    9 is one less than 10, and 11 is one more. So 9 times a number is 10 times it, minus the number once. And 11 times a number is 10 times it, plus the number once.

    6 rows of 9 are 6 rows of 10 with one square missing in each row: 60 − 6 = 54.

    For 11 times a two-digit number there is a faster way. Write the two digits apart and put their sum in the middle. If the sum is 10 or more, write only its last digit and carry the 1 to the left digit.

    1. 9 × 37
    2. 10 × 37 − 37
    3. 370 − 37
    4. 333
    1. 11 × 43
    2. 4 (4 + 3) 3
    3. 4 7 3
    4. 473
    The middle digit is the sum of the two digits.
    1. 11 × 76
    2. 7 (7 + 6) 6
    3. 7 13 6
    4. 8 3 6
    5. 836
    The sum is 13: write the 3 in the middle and carry the 1 to the 7.
    Next step: Double and halve
  5. 5

    Double one number, halve the other

    If you double one number (multiply it by 2) and halve the other (divide it by 2), the answer stays the same. Keep going until one number ends in 0.

    It works best when one number is even and the other ends in 5, because doubling a number that ends in 5 gives a number that ends in 0.

    1. 16 × 35
    2. 8 × 70
    3. 560
    Half of 16 is 8, and double 35 is 70. 8 × 7 = 56, so 8 × 70 = 560.
    1. 14 × 45
    2. 7 × 90
    3. 630
    7 × 9 = 63, so 7 × 90 = 630.
    Next step: Estimate first
  6. 6

    Estimate first, then rule out choices

    Before you calculate, round each number to an easy one and get a rough answer. The answer choices are often far apart, so the estimate alone can rule out some of them.

    Rounding to an easy number
    NumberRoundedWhy
    47507 is 5 or more, so round up
    412400it is 12 away from 400
    1920it is 1 away from 20
    $4.85$5it is 15 cents from $5

    Then check the last digit. The last digit of a product comes only from the last digits of the two numbers. In 412 × 19, it is the last digit of 2 × 9 = 18: an 8.

    Example

    Which is 412 × 19? 4,128 · 7,826 · 7,828 · 78,280

    1. Round: 412 is about 400, and 19 is about 20. 400 × 20 = 8,000.
    2. 4,128 and 78,280 are far from 8,000. Rule them out.
    3. Last digit: 2 × 9 = 18 ends in 8, so the answer ends in 8. That rules out 7,826.
    4. Only 7,828 is left, and you never did the full multiplication.

    Answer 7,828

    Next step: Three examples
  7. 7

    Three worked examples, from easy to test level

    Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.

    The test-level one uses both tools: first an estimate to rule out choices, then a mental shortcut to decide between the two that are left.

    Example · Easy

    What is 64 × 5?

    First try64 × 5 → 64 × 10 = 640

    1. ×5 is not ×10. 5 is half of 10, so the answer must be half of 640.
    2. 64 × 10 = 640.
    3. Half of 640: 640 ÷ 2 = 320.
    4. Check with an estimate: 60 × 5 = 300, close to 320.

    Answer 320

    Example · Medium

    A unit orders 24 boxes of field rations at $15 a box. What is the total cost?

    First try24 × 15 → 20 × 10 + 4 × 5 = 220

    1. Multiplying tens by tens and ones by ones misses two pieces: 20 × 5 = 100 and 4 × 10 = 40. That is why 220 is too small.
    2. Halve 24 and double 15: 24 × 15 is the same as 12 × 30.
    3. 12 × 3 = 36, so 12 × 30 = 360.
    4. Check by splitting only one number: 24 × 10 = 240 and 24 × 5 = 120. 240 + 120 = 360.

    Answer $360

    Example · Test level

    A supply sergeant buys 29 cases of water at $4.85 a case. What is the total? $116.00 · $140.65 · $145.50 · $1,406.50

    First tryRound $4.85 down to $4: 29 × 4 = 116, so $116.00

    1. Rounding $4.85 down to $4 throws away 85 cents on every case, about $25 in all. Round to the closest easy number instead: $5.
    2. Estimate: 29 is about 30 and $4.85 is about $5. 30 × 5 = 150. Both numbers went up, so the real total is a little under $150.
    3. $1,406.50 is ten times too big, and $116.00 is far below 150. Two choices are left: $140.65 and $145.50.
    4. Work it out: $4.85 is 15 cents less than $5. 30 cases at $5 cost 30 × 5 = 150. Each case was 15 cents too much: 30 × 0.15 = 4.50. So 30 cases cost 150 − 4.50 = 145.50.
    5. But there are 29 cases, not 30. Take one case back: 145.50 − 4.85 = 140.65.

    Answer $140.65

Next: Watch out

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.

Trap

Changing only one number

In subtraction, rounding one number and forgetting the other. 72 − 29:

Wrong72 − 30 = 42

RightAdd 1 to both: 73 − 30 = 43

Trap

Forgetting to halve

×5 done as ×10, with no half at the end. 86 × 5:

Wrong86 × 10 = 860

Right860 ÷ 2 = 430

Trap

Doubling both numbers

Double one number and halve the other. Doubling both makes the answer four times too big. 18 × 25:

Wrong36 × 50 = 1,800

Right9 × 50 = 450

Trap

Trusting the closest choice

An estimate rules out far choices, not close ones. 61 × 39, with choices 2,379 and 2,400:

Wrong60 × 40 = 2,400, so 2,400

Right1 × 9 = 9: it ends in 9, so 2,379

Next: Practice

15 questions in three levels

Warm-up first, then core, then test level. Pick an answer, then read why.

On test day

Mental math is not a test of its own. You use it on every math question, in Arithmetic Reasoning (word problems) and Mathematics Knowledge. On the computer test, AR has 15 questions in 55 minutes and MK has 15 questions in 31 minutes. On paper, AR is 30 questions in 36 minutes and MK is 25 questions in 24 minutes.

There is no calculator. Before you work anything out, estimate and look at the answer choices. A choice that is far from your estimate, or that has the wrong last digit, can be ruled out in seconds.

On the computer test you cannot go back to a question. If you are stuck, rule out what you can with your estimate, choose from what is left, and move on.

Question 1 of 15 · Warm-up

What is 67 + 29?

Show the answer

B. 29 is one less than 30. 67 + 30 = 97, then take back the 1: 97 − 1 = 96. 97 forgets to take the 1 back, 98 adds it instead, and 86 forgets to carry the ten from 7 + 9 = 16. The answer is 96.

Question 2 of 15 · Warm-up

What is 84 × 5?

Show the answer

D. Times 10, then half: 84 × 10 = 840, and half of 840 is 420. 400 is only 80 × 5 and forgets the 4 × 5. 210 takes half twice, and 89 adds 5 instead of multiplying. 84 × 5 is 420.

Question 3 of 15 · Warm-up

What is 34 × 11?

Show the answer

C. Write the digits apart, 3 and 4, and put their sum in the middle: 3 + 4 = 7, so the answer is 374. Check: 340 + 34 = 374. 340 is only 34 × 10, 347 puts the 7 at the end, and 3,434 writes 34 twice. 34 × 11 is 374.

Question 4 of 15 · Warm-up

What is 8 × 99?

Show the answer

A. 99 is one less than 100. 8 × 100 = 800, but that is one 8 too many, so take it back: 800 − 8 = 792. 799 takes back 1 instead of 8, 808 adds the 8, and 800 stops after rounding. 8 × 99 is 792.

Question 5 of 15 · Warm-up

Which is the best estimate of 6.1 × 49?

Show the answer

B. Round each number to the closest easy one: 6.1 is about 6, and 49 is about 50. 6 × 50 = 300. 350 rounds 6.1 up to 7, which is farther away. 30 and 3,000 put the decimal point in the wrong place. The best estimate is 300.

Question 6 of 15 · Core

A box holds 25 water bottles. How many bottles are in 16 boxes?

Show the answer

B. 25 is a quarter of 100. A quarter of 16 is 4, and 4 × 100 = 400. Or: 16 × 100 = 1,600, and halving twice gives 800, then 400. 1,600 forgets to divide, 800 halves only once, and 320 is 16 × 20, which drops the 16 × 5. The number of bottles is 400.

Question 7 of 15 · Core

What is 35 × 18?

Show the answer

C. Double 35 and halve 18: 35 × 18 is the same as 70 × 9. 7 × 9 = 63, so 70 × 9 = 630. 1,260 doubles 35 but forgets to halve 18, 315 halves 18 but forgets to double 35, and 540 is 30 × 18, which drops the 5 × 18. 35 × 18 is 630.

Question 8 of 15 · Core

What is 9 × 47?

Show the answer

A. 9 × 47 is 10 × 47 minus one 47: 470 − 47. Make a ten: add 3 to both numbers, and 473 − 50 = 423. 433 borrows for the ones (10 − 7 = 3) but forgets to take 1 from the 7 tens, 463 takes away only 7, and 517 adds the 47 instead, which is 11 × 47. 9 × 47 is 423.

Question 9 of 15 · Core

What is 1,000 − 386?

Show the answer

A. Move both numbers by the same amount, as in Make a ten: take 1 away from both. The gap does not change: 999 − 385. Now no column needs borrowing: 9 − 5 = 4, 9 − 8 = 1, 9 − 3 = 6, so 614. 714 forgets that the hundreds column lent 1 and does 10 − 3 = 7 there, 724 takes every digit away from 10, and 1,386 subtracts the smaller digit from the bigger one in each column. 1,000 − 386 is 614.

Question 10 of 15 · Core

What is 6 × 73?

Show the answer

C. Split 73 into tens and ones: 70 + 3. Multiply each part by 6: 6 × 70 = 420 and 6 × 3 = 18. Add them: 420 + 18 = 438. 420 is only the 6 × 70 and drops the 6 × 3, 428 writes the 8 of 18 but forgets to carry its 1 ten, and 4,218 writes 42 and 18 side by side instead of adding them. 6 × 73 is 438.

Question 11 of 15 · Test level

A unit buys 25 folding chairs at $36 each and one table for $95. What is the total cost?

Show the answer

B. Chairs first. 25 is a quarter of 100, and a quarter of 36 is 9, so the chairs cost 9 × 100 = 900 dollars. Then add the table: 900 + 95 = 995. $900 forgets the table, $3,695 forgets to take the quarter (3,600 + 95), and $1,895 takes half instead (1,800 + 95). The total is $995.

Question 12 of 15 · Test level

A ship’s store sells 198 phone cards at $9.95 each. About how much money is that?

Show the answer

C. Round: 198 is about 200, and $9.95 is about $10. 200 × 10 = 2,000. $1,800 rounds $9.95 down to $9, but $9.95 is only 5 cents from $10. $20,000 adds an extra zero, and $200 loses one. It is about $2,000.

Question 13 of 15 · Test level

A recipe for 4 people uses 3 cups of rice. The dining facility cooks for 96 people. How many cups of rice does it need?

Show the answer

D. First, how many times the recipe: 96 ÷ 4. Dividing by 4 is halving twice: 96, then 48, then 24. So it is 24 recipes, and each uses 3 cups: 24 × 3 = 72. 24 stops after the first step, 48 is 96 ÷ 2: it halves only once and forgets the 3 cups, and 32 divides 96 by the 3 cups instead of by the 4 people. It needs 72.

Question 14 of 15 · Test level

Eleven soldiers each pay $68 for a group trip. How much do they pay in all?

Show the answer

D. This is 68 × 11. Write the digits apart, 6 and 8, and put their sum in the middle: 6 + 8 = 14. The sum is more than 9, so write the 4 and carry the 1 to the 6: 748. Check: 680 + 68 = 748. $648 forgets the carry, $680 pays for only 10 soldiers, and $612 takes one 68 away instead of adding it, which is 9 × 68. They pay $748.

Question 15 of 15 · Test level

A battalion orders 52 crates of bottled water. Each crate weighs 38 pounds. What is the total weight?

Show the answer

A. Estimate: 50 × 40 = 2,000, so 19,760 pounds is far too big. Last digit: 2 × 8 = 16, so the answer ends in 6, which rules out 2,000 and 2,078. To be sure: 52 × 4 = 208, so 52 × 40 = 2,080. That is 52 × 2 = 104 too much. 2,080 − 104 = 1,976. 2,000 is the estimate, not the answer, and 2,078 takes back 2 instead of 104. The total is 1,976 pounds.

Next: Flashcards

12 flashcards

Tap a card to flip it. Keep the ones you miss until they stick.

12 cards
Lesson complete

You finished this lesson: Mental math and estimating.

Practice score: not taken yet Take the practice

Next topic: Fractions Course map
In the app

See 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.

Try the app free