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State Board · Class 5 · Mathematics

Lesson 4: Multiplication and Division

5 concepts · 40 practice questions with answers · Free to read

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Multiplying by a three-digit number

From daily life

A grocer packs 46 tins of oil in every box and sends 213 boxes to a town. To find all the tins, adding 46 again and again 213 times would take too long, so we multiply.

Simple definition

In 46 × 213, 46 is the multiplicand (गुण्य), 213 is the multiplier (गुणक) and 9,798 is the product (गुणाकार). Multiply by the ones, the tens and the hundreds of the multiplier separately and add the three partial products.

Multiplication in steps: 754 × 368 = 2,77,472, with the partial products shown.754368×6,032754 × 845,240754 × 602,26,200754 × 3002,77,472add all

In the figure : Multiplying by a three-digit number: multiply by the ones, tens and hundreds separately and add the three products.

Rule 1: Three partial products — Ones: × 3. Tens: × 10. Hundreds: × 200. Then add.

  • Easy : 46 × 213: 46 × 3 = 138; 46 × 10 = 460; 46 × 200 = 9,200; total 9,798.
  • Medium : 327 × 145: 327 × 5 = 1,635; 327 × 40 = 13,080; 327 × 100 = 32,700; total 47,415.
  • Exam-type : 608 × 207: 608 × 7 = 4,256; tens digit is 0 so that line is 0; 608 × 200 = 1,21,600; total 1,25,856.
Live figure

Multiplying by a three-digit number

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Rule 2: Zero in the multiplier — A zero digit gives a zero partial product. Do not skip the place.

  • Easy : 12 × 101 = 12 + 0 + 1,200 = 1,212.
  • Medium : 35 × 120 = 35 × 12 × 10 = 420 × 10 = 4,200.
  • Exam-type : 508 × 230 = 508 × 23 × 10 = 11,684 × 10 = 1,16,840.
Remember

Product = multiplicand × multiplier. Add the partial products in the right places.

Common mistake

Writing every partial product starting under the ones place. The tens line starts one place left, the hundreds line two places left.

Shortcut

Memory clue: Ones, Tens, Hundreds: shift left each time — add a zero for the tens line and two zeros for the hundreds line.

Spot the difference

Multiplier digitMultiply by
ones (3)× 3
tens (1)× 10
hundreds (2)× 200

Practice questions

Easy
  1. 1Name the multiplicand and the multiplier in 25 × 134.
  2. 2Find: 12 × 101.
  3. 3Find: 35 × 120.
Medium
  1. 4Find: 214 × 213.
  2. 5Find: 346 × 105.
  3. 6Find: 508 × 230.
Challenging
  1. 7Find: 1,250 × 312.
  2. 8A shopkeeper buys 148 sets of books at 275 rupees each. How much does he pay?
Show answers (try them yourself first)
  1. 1Multiplicand 25; multiplier 134.
  2. 21,212
  3. 34,200
  4. 445,582
  5. 536,330
  6. 61,16,840
  7. 73,90,000
  8. 8148 × 275 = 40,700 rupees.

Multiplication word problems and pairing off objects

From daily life

A school ordered notebooks: 36 boxes with 144 notebooks in each box. Another day a shirt shop owner wants to know how many different dress combinations 4 shirts and 3 caps can make. Both are multiplication.

Simple definition

In multiplication word problems, words like "each", "every" and "in all" show equal groups. When two groups are paired off, the number of different pairs is the product of the numbers in the two groups.

3 shirts and 2 pairs of trousers make 6 different pairs (3 × 2).each shirt with each pair of trousersTrousers 1Trousers 2Shirt 1Shirt 2Shirt 33 × 2 = 6 pairs

In the figure : Pairing off: 3 shirts and 2 pairs of trousers give 3 × 2 = 6 different pairs.

Rule 1: Equal groups — Number of groups × number in each group = total.

  • Easy : 36 boxes × 144 notebooks = 5,184 notebooks.
  • Medium : 125 rows × 36 plants in each row = 4,500 plants.
  • Exam-type : 2,350 rupees per month × 18 months = 42,300 rupees; 4,500 × 24 = 1,08,000 rupees.

Rule 2: Pairing off — Pairs = size of the first group × size of the second group.

  • Easy : 4 shirts and 3 caps give 4 × 3 = 12 pairs.
  • Medium : 5 pens and 6 notebooks give 5 × 6 = 30 pairs.
  • Exam-type : 7 boys and 4 hats give 7 × 4 = 28 photographs, one for each boy-hat pair.
Remember

Pairs from two groups = product of the group sizes (not the sum).

Common mistake

Adding 3 + 2 = 5 pairs. Every item of one group goes with every item of the other, so 3 × 2 = 6.

Shortcut

Memory clue: "Each" means multiply.

Spot the difference

WordsMeaning
each / everymultiply
pairs of two groupsmultiply
total of equal groupsmultiply

Practice questions

Easy
  1. 1Find: 36 × 144 notebooks.
  2. 2How many pairs can 4 shirts and 3 caps make?
  3. 3A garden has 125 rows with 36 plants in each row. How many plants are there?
Medium
  1. 45 pens and 6 notebooks: how many pairs?
  2. 5A bus carries 45 passengers. How many passengers do 120 buses carry?
  3. 6Aarti saves 850 rupees every month. How much does she save in 26 months?
Challenging
  1. 7A canteen has 7 kinds of sandwiches and 4 kinds of drinks. How many different meal pairs are possible?
  2. 8A factory makes 2,350 bulbs a day. How many bulbs does it make in 18 days, and how many in 24 days?
Show answers (try them yourself first)
  1. 15,184
  2. 212
  3. 34,500
  4. 430
  5. 545 × 120 = 5,400
  6. 6850 × 26 = 22,100 rupees.
  7. 77 × 4 = 28 pairs.
  8. 818 days: 2,350 × 18 = 42,300. 24 days: 2,350 × 24 = 56,400.

Dividing by one-digit and two-digit numbers

From daily life

Fifty-four mangoes are shared equally among 6 children. Each child gets 54 ÷ 6 = 9 mangoes. Division makes equal parts.

Simple definition

Division means making equal parts or subtracting repeatedly. In a division, the dividend (भाज्य) is divided by the divisor (भाजक) to get the quotient (भागाकार) and the remainder (बाकी). Check: dividend = divisor × quotient + remainder.

Long division of 4925 by 7: quotient 703 and remainder 4.7037492594−020−2512−4quotientremainder = 4

In the figure : Long division of 4,925 ÷ 7: 49 hundreds ÷ 7 = 7, the tens place gets 0; quotient 703, remainder 4.

Rule 1: Divide place by place — Start from the highest place. If it is too small to divide, take the next digit as well.

  • Easy : 72 ÷ 6 = 12 because 6 × 12 = 72.
  • Medium : 3,456 ÷ 8 = 432.
  • Exam-type : 6,058 ÷ 9 = quotient 673, remainder 1. The tens place is 7: 9 × 673 = 6,057 and 6,058 − 6,057 = 1.
The division check: dividend 4925 = divisor 7 × quotient 703 + remainder 4.4925dividend=7divisor×703quotient+4remainder

In the figure : Check: dividend = divisor × quotient + remainder, so 4,925 = 7 × 703 + 4.

Rule 2: Two-digit divisor — Work in the same way. Make the table of the divisor first if needed.

  • Easy : 7,500 ÷ 25 = 300.
  • Medium : 5,432 ÷ 24 = quotient 226, remainder 8.
  • Exam-type : 8,165 ÷ 32 = quotient 255, remainder 5; check 32 × 255 + 5 = 8,160 + 5 = 8,165.
Remember

The remainder is always smaller than the divisor.

Common mistake

Forgetting to write 0 in the quotient when a place cannot be divided. In 4,925 ÷ 7 the tens place has quotient 0, giving 703, not 73.

Shortcut

Memory clue: D-M-S-B — Divide, Multiply, Subtract, Bring down.

Spot the difference

TermMeaningIn 17 ÷ 5 = 3 r 2
dividendnumber being divided17
divisornumber we divide by5
quotienthow many times3
remainderwhat is left2

Practice questions

Easy
  1. 1Write the two divisions that come from 9 × 4 = 36.
  2. 2Divide: 72 ÷ 6.
  3. 3Divide: 354 ÷ 6.
Medium
  1. 4Divide: 3,456 ÷ 8.
  2. 5Divide: 7,500 ÷ 25.
  3. 6Divide 5,432 by 24 and give the remainder.
Challenging
  1. 7Divide 6,058 by 9 and check the answer.
  2. 8Divide 8,165 by 32 and check.
Show answers (try them yourself first)
  1. 136 ÷ 9 = 4 and 36 ÷ 4 = 9.
  2. 212
  3. 359
  4. 4432
  5. 5300
  6. 6Quotient 226, remainder 8.
  7. 7Quotient 673, remainder 1. Check: 9 × 673 + 1 = 6,058.
  8. 8Quotient 255, remainder 5. Check: 32 × 255 + 5 = 8,165.

Guessing the quotient and division word problems

From daily life

A shopkeeper buys 35 kg of rice for 910 rupees. To know the rate of 1 kg, divide 910 by 35. Guessing the quotient makes such division quick.

Simple definition

To guess the quotient, look at the first digits of the dividend and the divisor, guess a digit and check by multiplying. If the product is bigger than what you have, try a smaller digit.

Table of 45: 45 × 1 = 45, 45 × 2 = 90, 45 × 3 = 135, 45 × 4 = 180, 45 × 5 = 225, 45 × 6 = 270, 45 × 7 = 315, 45 × 8 = 360, 45 × 9 = 405Multiples of 4545 × 1 = 4545 × 2 = 9045 × 3 = 13545 × 4 = 18045 × 5 = 22545 × 6 = 27045 × 7 = 31545 × 8 = 36045 × 9 = 405

In the figure : Guessing the quotient of 9,842 ÷ 45: 45 × 2 = 90 < 98; 45 × 8 = 360 < 392 but 45 × 9 = 405 > 392.

Rule 1: Guess and check — Choose the biggest digit whose product is not bigger than the part you are dividing.

  • Easy : 7,684 ÷ 36: 36 × 2 = 72 ≤ 76, so the hundreds digit is 2.
  • Medium : Remainder 4 hundreds and 8 tens = 48 tens; 36 × 1 = 36 ≤ 48, so the tens digit is 1.
  • Exam-type : Remainder 12 tens and 4 ones = 124; 36 × 3 = 108 ≤ 124 but 36 × 4 = 144 > 124, so the ones digit is 3. Quotient 213, remainder 16.
Long division of 9842 by 45: quotient 218 and remainder 32.21845984209−8454−932063−23quotientremainder = 32

In the figure : Long division of 9,842 ÷ 45: the digits 2, 1 and 8 of the quotient come from the guesses; quotient 218, remainder 32.

Rule 2: Word problems — Words like "each", "per", "equal parts" and "how many times" tell us to divide.

  • Easy : Total cost ÷ weight = rate: 1,260 ÷ 28 = 45 rupees per kg.
  • Medium : 2,730 books shared equally among 35 libraries: 2,730 ÷ 35 = 78 books each.
  • Exam-type : A rope of 9,000 cm is cut into pieces of 45 cm: 9,000 ÷ 45 = 200 pieces.
Remember

Always check: divisor × quotient + remainder = dividend.

Common mistake

Guessing a digit that is too big and not correcting it. If the product is too big, use the next smaller digit.

Shortcut

Memory clue: Round the divisor (like 36 as about 40) to guess faster, then confirm.

Spot the difference

StepWhat to do
1Look at the first digits
2Guess a digit and multiply
3Subtract and check

Practice questions

Easy
  1. 1Divide: 1,260 ÷ 28.
  2. 2Divide: 2,730 ÷ 35.
  3. 3Divide: 4,500 ÷ 18.
Medium
  1. 4Divide: 9,000 ÷ 45.
  2. 5Divide 7,684 by 36. Give the quotient and the remainder.
  3. 6A shopkeeper pays 910 rupees for 35 kg of rice. Find the rate per kg.
Challenging
  1. 73,000 oranges are packed 40 in a box. How many boxes are needed?
  2. 8A tailor has 4,590 cm of cloth. How many 17 cm pieces can he cut, and is anything left?
Show answers (try them yourself first)
  1. 145
  2. 278
  3. 3250
  4. 4200
  5. 5Quotient 213, remainder 16.
  6. 6910 ÷ 35 = 26 rupees per kg.
  7. 73,000 ÷ 40 = 75 boxes.
  8. 84,590 ÷ 17 = 270 pieces, nothing left.

Mixed examples

From daily life

A farmer plants 120 trays of seedlings with 25 seedlings in each tray, and later plants them in rows of 40. First multiply, then divide.

Simple definition

Many problems need two steps. Read the problem, decide the operation for each step (multiply for equal groups, divide for sharing) and always check the answer.

Rule 1: Two-step problems — Do one step at a time and write the answer of the first step before going to the next.

  • Easy : 120 trays × 25 seedlings = 3,000 seedlings; 3,000 ÷ 40 = 75 rows.
  • Medium : 215 pencils per class × 48 classes = 10,320 pencils; shared equally among 24 schools gives 10,320 ÷ 24 = 430 each.
  • Exam-type : A hostel buys 8 sacks of 98 kg rice = 784 kg. For 14 families that is 784 ÷ 14 = 56 kg each.

Rule 2: Money problems — Total spent, total earned and profit or loss need careful steps.

  • Easy : 8 pens at 12 rupees each cost 8 × 12 = 96 rupees.
  • Medium : 260 bags at 50 rupees = 13,000; if expenses are 11,450 the profit is 13,000 − 11,450 = 1,550.
  • Exam-type : A tank holds 3,000 litres. 16 houses need 225 litres each, which is 3,600 litres. The tank falls short by 3,600 − 3,000 = 600 litres.
Remember

Estimate first: if 100 × 100 = 10,000, then 98 × 102 should be close to 10,000.

Common mistake

Mixing up the order: dividing before multiplying. Read what the question asks in each step.

Shortcut

Memory clue: Read, Choose, Calculate, Check — R-C-C-C.

Spot the difference

WordsOperation
each / in allmultiply
shared equallydivide
profit / losssubtract

Practice questions

Easy
  1. 1Find: 120 × 25.
  2. 2Find: 3,000 ÷ 40.
  3. 3A box has 24 pencils. How many pencils are in 125 boxes?
Medium
  1. 43,000 pencils are shared equally among 50 children. How many does each child get?
  2. 58 sacks of rice weigh 98 kg each. How many kilograms in all?
  3. 6The 784 kg in the question above is shared equally among 14 families. How much does each family get?
Challenging
  1. 7A trader buys 260 bags at 50 rupees each and spends 11,450 rupees. Find his profit.
  2. 8A tank holds 3,000 litres. 16 houses need 225 litres each. Is the water enough? If not, by how much is it short?
Show answers (try them yourself first)
  1. 13,000
  2. 275
  3. 324 × 125 = 3,000
  4. 43,000 ÷ 50 = 60
  5. 58 × 98 = 784 kg
  6. 6784 ÷ 14 = 56 kg.
  7. 7260 × 50 = 13,000; profit = 13,000 − 11,450 = 1,550 rupees.
  8. 816 × 225 = 3,600 litres needed; the tank is short by 600 litres.

Quick revision

  • Multiplicand × multiplier = product; pairs from two groups = product of the group sizes.
  • Dividend = divisor × quotient + remainder.
  • Guess the quotient digit from the first digits and check by multiplying.