Scaling both numbers by the same amount leaves the answer completely unchanged
How many fit?
6 ÷ 0.4 cannot mean sharing 6 among 0.4 of a group. But division has a second meaning: how many times does one number fit into the other? A ribbon 6 m long is cut into pieces 0.4 m long. How many pieces are there? That count is 6 ÷ 0.4.
Lay copies of 0.4 along 6 and count them: 15 copies fit exactly. So 6 ÷ 0.4 = 15.
Make the divisor 0.4, then scale both bars by 10
The long bar is 6. Shrink the gold divisor to 0.4 and count the copies that fit along the bar: 15. Then choose × 10. Both numbers are now ten times as big, 60 and 4, and the same 15 copies fit.
Make the divisor a whole number
Counting copies one at a time is slow. Instead, turn the divisor into a whole number. Multiply both numbers by 10: every digit moves one place to the left, and the decimal point stays where it is. 6 becomes 60, and 0.4 becomes 4.
Multiplying both by 10 makes the whole ten times as long and every piece ten times as long, so the same number of pieces fits. 6 ÷ 0.4 has the same answer as 60 ÷ 4.
Both numbers are multiplied by 10: the digits of 6 and of 0.4 each move one place to the left, so 6 becomes 60 and 0.4 becomes 4.
Then divide as usual
60 ÷ 4 is ordinary division. 4 goes into 6 once, with 2 left over. Bring down the 0 to make 20, and 4 goes into 20 five times. So 60 ÷ 4 = 15, and 6 ÷ 0.4 = 15 too.
Check by multiplying back. 15 × 4 = 60, and 0.4 has one decimal place, so 15 × 0.4 = 6.0, which is 6.
60 ÷ 4 = 15, so 6 ÷ 0.4 = 15.
Two decimal places, and a point that stays
When the divisor has two decimal places, multiply by 10 twice, which is multiplying by 100. For 3 ÷ 0.25, multiplying both by 10 gives 30 ÷ 2.5, and the divisor still has a point. Multiply by 10 again: 300 ÷ 25 = 12. That makes sense: 0.25 is a quarter, four quarters fit into each whole, and 3 wholes hold 12 of them.
Only the divisor has to become whole. The number being divided may keep its point. For 15.6 ÷ 0.4, multiply both by 10 to get 156 ÷ 4, which is 39.
A bigger answer is right
The answer 15 is bigger than 6, the number that was divided. That is correct. 0.4 is less than 1, and many small pieces fit into 6. Dividing by a number less than 1 always gives an answer bigger than the number you started with.
The usual mistake is to multiply only one of the numbers by 10. Dividing 60 by 0.4 gives 150, ten times too big. The two numbers are scaled together, by the same amount, or the count changes.
Decimals · Cutting Equal Pieces with a Remainder
QuestionA rope 9.2 m long is cut into pieces that are each 0.75 m long. (a) What is the greatest number of complete pieces that can be cut? (b) What length of rope is left over?
1.Divide the rope by the length of one piece: 9.2 ÷ 0.75. Multiply both numbers by 100 so that the divisor is a whole number: 920 ÷ 75.
9.2 ÷ 0.75 is the same as 920 ÷ 75. 2.75 × 12 = 900 and 75 × 13 = 975, which is more than 920. So only 12 complete pieces fit.
(a) 75 × 12 = 900 fits in 920 and 75 × 13 = 975 does not: 12 pieces. 3.(a) The greatest number of complete pieces is 12.
4.The 12 pieces use 12 × 0.75 = 9 m of rope.
The 12 pieces use 12 × 0.75 = 9 m. 5.(b) The length left over is 9.2 − 9 = 0.2 m.
(b) 9.2 − 9 = 0.2 m is left over.
Answer: (a) 12 pieces; (b) 0.2 m
Watch out for
- Rounding 12.26… up to 13 pieces. The thirteenth piece would be shorter than 0.75 m, so it is not a complete piece.
- Reading the remainder of 920 ÷ 75, which is 20, as 20 m or 2 m. Both numbers were multiplied by 100, so the remainder is 20 hundredths of a meter, which is 0.2 m.
Decimals · Comparing Two Pack Sizes by Unit Price
QuestionPack A holds 0.75 kg of almonds and costs $4.20. Pack B holds 1.2 kg of the same almonds and costs $6.60. (a) What is the price of 1 kg of almonds in each pack? (b) Mrs Lim needs exactly 6 kg of almonds. How much does she save by buying only the cheaper kind of pack?
1.Find the price of 1 kg in Pack A: 4.20 ÷ 0.75 = 420 ÷ 75 = $5.60.
Pack A: 4.20 ÷ 0.75 = $5.60 for 1 kg. 2.Find the price of 1 kg in Pack B: 6.60 ÷ 1.2 = 66 ÷ 12 = $5.50.
Pack B: 6.60 ÷ 1.2 = $5.50 for 1 kg. 3.(a) Pack A costs $5.60 per kilogram and Pack B costs $5.50 per kilogram, so Pack B is cheaper.
(a) $5.60 and $5.50 per kilogram. Pack B is cheaper. 4.For 6 kg she needs 6 ÷ 0.75 = 8 of Pack A, costing 8 × 4.20 = $33.60, or 6 ÷ 1.2 = 5 of Pack B, costing 5 × 6.60 = $33.
Six kilograms is 8 of Pack A or 5 of Pack B. 5.(b) She saves 33.60 − 33 = $0.60.
(b) She saves $0.60.
Answer: (a) Pack A: $5.60 per kg, Pack B: $5.50 per kg; (b) $0.60
Watch out for
- Choosing Pack A because $4.20 is less than $6.60. The packs hold different masses, so compare the price of 1 kg.
- Dividing the mass by the cost (0.75 ÷ 4.20). That gives the kilograms bought with one dollar, not the price of one kilogram.
Your turn
1.15.6 ÷ 0.4