Multiplying and dividing by hand

Multiplying is equal groups; dividing is sharing

How to solve them

1. 1

   Break the big number into hundreds, tens and ones

   To multiply a big number by a one-digit number, break the big number into its place values: hundreds, tens and ones. Multiply each piece by the small number. Then add the pieces.

   1. 7 × 346
   2. 7 × (300 + 40 + 6)
   3. 7 × 300 + 7 × 40 + 7 × 6
   4. 2,100 + 280 + 42
   5. 2,422

   × 10

   Write one zero at the end.

   36 × 10 = 360

   × 100

   Write two zeros at the end.

   36 × 100 = 3,600

   × 30

   Multiply by 3, then write one zero.

   36 × 30 = 1,080

   Shortcut

   Zeros wait at the end

   When a number ends in zero, hide the zero, multiply what is left, and put the zero back. 40 × 23: first 4 × 23 = 92, then put the zero back: 920.

   Next step: Two rows ↓
2. 2

   Two digits times two digits: two rows, then add

   When both numbers have two digits, split the second number into tens and ones. You get two rows, called partial products: the first number times the ones, and the first number times the tens. Then you add the two rows.

   1. 58 × 26
   2. 58 × (20 + 6)
   3. 58 × 20 + 58 × 6
   4. 1,160 + 348
   5. 1,508

   The tens row is where people slip. 58 × 20 is 58 × 2 = 116 with a zero at the end: 1,160. Write that zero first, so the row moves one place to the left.

   The written way: 58 × 26

   Row	Work	Result
   Ones row	58 × 6	348
   Tens row	58 × 20	1,160
   Add the rows	348 + 1,160	1,508

   Shortcut

   Estimate first

   58 × 26 is close to 60 × 25 = 1,500. If your answer is near 150 or 15,000, a zero went missing or was added. Fix it before you look at the choices.

   Next step: Long division ↓
3. 3

   Long division: divide, multiply, subtract, bring down

   Long division works one digit at a time, from left to right. For each digit you make the same four moves, and then you start again with the next digit.

   The four moves, on the example 936 ÷ 4:

   Divide÷How many 4s fit in 9? Two.

   Multiply×2 × 4 = 8

   Subtract−9 − 8 = 1

   Bring down↓The next digit, 3, joins the 1 to make 13.

   With a two-digit divisor, round it to estimate each digit. In 192 ÷ 32, think of 32 as 30: 30 × 6 = 180, close to 192, so try 6. 6 × 32 = 192 exactly. If the product is too big, try one less.

   Example

   936 ÷ 4

   1. 9 ÷ 4: 2 fits, because 2 × 4 = 8. Subtract: 9 − 8 = 1. Write 2 on top.
   2. Bring down the 3 to make 13. 3 fits, because 3 × 4 = 12. Subtract: 13 − 12 = 1. Write 3 on top.
   3. Bring down the 6 to make 16. 16 ÷ 4 = 4 exactly. Write 4 on top.
   4. The digits on top read 234.

   Answer 234

   Shortcut

   When the divisor does not fit, write 0

   In 1,236 ÷ 6: 12 ÷ 6 = 2. Then 6 does not fit into 3, so write 0 on top and bring down the 6. 36 ÷ 6 = 6. The answer is 206, not 26.

   Shortcut

   Zeros at the end of both numbers

   When both numbers end in zero, cross out the same number of zeros from each. 2,400 ÷ 60 is the same as 240 ÷ 6 = 40. Check: 40 × 60 = 2,400.

   Next step: Remainders ↓
4. 4

   When it does not come out even: the remainder

   Sometimes the last subtraction does not reach zero. What is left over is the remainder, written with an R. A remainder is always smaller than the divisor. If it is not, one more fits.

   1. 587 ÷ 6
   2. 587 ÷ 6 → 9 on top, 58 − 54 = 4 left
   3. 47 ÷ 6 → 7 on top, 47 − 42 = 5 left
   4. 97 R 5

   Round up

   587 guests sit 6 to a table. How many tables?

   97 full tables, and 5 people still need a seat: 98

   Drop it

   587 eggs go in boxes of 6. How many full boxes?

   The 5 left over do not fill a box: 97

   The remainder is the answer

   587 pens are shared by 6 people, the same number each. How many are left?

   5

   Next step: Decimals ↓
5. 5

   A decimal in the dividend: the point goes straight up

   When the number you divide has a decimal point, put the point in the answer first, directly above the point in the dividend. Then divide as usual, as if the point were not there.

   If you run out of digits and something is still left, write a decimal point and zeros after the number and keep going. The zeros do not change its value: 7 and 7.00 are the same amount.

   Example

   $45.60 shared by 8 people

   1. Put the point on top first, right above the point in 45.60.
   2. 45 ÷ 8: 5 fits, because 5 × 8 = 40. Subtract: 45 − 40 = 5.
   3. Bring down the 6 to make 56. 56 ÷ 8 = 7 exactly.
   4. Bring down the 0: 0 ÷ 8 = 0. The top reads 5.70, so each person pays $5.70.

   Answer $5.70

   Example

   Out of digits: 7 ÷ 4

   1. 4 fits into 7 once, with 3 left.
   2. Write 7 as 7.00 and put the point on top. Bring down a 0 to make 30. 4 fits into 30 seven times, because 7 × 4 = 28, and 2 is left. Write 7 after the point.
   3. Bring down the other 0 to make 20. 20 ÷ 4 = 5 exactly.
   4. The answer is 1.75.

   Answer 1.75

   Next step: Check it ↓
6. 6

   Check any division by multiplying back

   quotient × divisor + remainder = dividend

   Multiplying undoes dividing. So after a division, multiply your answer by the divisor and add the remainder. If you get the number you started with, the division is right. It is faster than doing the division again, and it catches the missing zero.

   Example

   Is 587 ÷ 6 really 97 R 5?

   1. Multiply back: 97 × 6. Break it apart: 90 × 6 = 540 and 7 × 6 = 42, so 540 + 42 = 582.
   2. Add the remainder: 582 + 5 = 587.
   3. That is the number we started with, so the division is right.

   Answer Correct

   Example

   Is 1,236 ÷ 6 equal to 26?

   1. Multiply back: 26 × 6 = 156. That is nowhere near 1,236, so it is wrong.
   2. A zero is missing on top. 206 × 6 = 1,236, so the answer is 206.

   Answer 206

   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 is a word problem in two steps: first a division with a remainder, then a multiplication. Decide what the remainder means before you take the second step.

   Example · Easy

   What is 8 × 265?

   First try8 × 200 + 8 × 60 + 5 = 2,085

   1. The 5 was added, not multiplied. Every piece of 265 must be multiplied by 8.
   2. 8 × 200 = 1,600 and 8 × 60 = 480.
   3. 8 × 5 = 40.
   4. Add the pieces: 1,600 + 480 + 40 = 2,120.

   Answer 2,120

   Example · Medium

   What is 2,135 ÷ 7?

   First try21 ÷ 7 is 3, then 35 ÷ 7 is 5, so the answer is 35

   1. 21 ÷ 7 = 3. Write 3 on top. Subtract: 21 − 21 = 0.
   2. Bring down the 3. 7 does not fit into 3, so write 0 on top. This is the step the first try skipped.
   3. Bring down the 5 to make 35. 35 ÷ 7 = 5. Write 5 on top.
   4. The top reads 305. Check: 305 × 7 = 2,135. The answer is 305.

   Answer 305

   Example · Test level

   A dining facility needs 940 eggs for a week. Eggs come in cartons of 12, and each carton costs $3. How much do the eggs cost?

   First try940 ÷ 12 is 78 R 4, so 78 × 3 = $234

   1. Cartons: 940 ÷ 12. 12 into 94: 7 fits, because 7 × 12 = 84, and 94 − 84 = 10. Bring down the 0 to make 100. 8 fits, because 8 × 12 = 96, and 100 − 96 = 4. So 78 R 4.
   2. 78 cartons hold 78 × 12 = 936 eggs. The 4 eggs left over need one more carton: 79 cartons.
   3. Cost: 79 × 3. Think of 80 × 3 = 240, then take away one 3: 240 − 3 = 237.
   4. The eggs cost $237.

   Answer $237

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.

15 questions in three levels

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

12 flashcards

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