Applications: Decimals

Draw one bar for each ribbon. The red bar is the blue bar with 1.8 m more on the end, so taking that piece away leaves two equal bars.

1. Draw the blue ribbon as one bar. The red ribbon is the same bar with an extra 1.8 m on the end.
2. Take the extra 1.8 m away from the total: 7.4 − 1.8 = 5.6 m.
3. What is left is two equal bars, so one bar is 5.6 ÷ 2 = 2.8 m.
4. (a) The blue ribbon is 2.8 m long.
5. (b) The red ribbon is 2.8 + 1.8 = 4.6 m long. Check: 2.8 + 4.6 = 7.4.

answer(a) 2.8 m; (b) 4.6 m

techniqueSubtracting in Columns · Dividing a Decimal

examsPSLE · GCSE Higher

Draw the original number as 1 unit. The new number is 10 times as large, so it is 10 units, and the difference between them is 9 units.

1. Moving the decimal point one place to the right multiplies a number by 10. Draw the original number as 1 unit and the new number as 10 units.
2. The new number is 10 − 1 = 9 units more than the original, so 9 units = 23.85.
3. 1 unit = 23.85 ÷ 9 = 2.65.
4. (a) The original number is 2.65.
5. (b) The new number is 2.65 × 10 = 26.5. Check: 26.5 − 2.65 = 23.85.

answer(a) 2.65; (b) 26.5

techniqueMultiplying and Dividing Decimals by 10, 100, 1,000 · Dividing a Decimal

examsPSLE

Draw the two purchases one under the other. The three pens are the same in both, so the second bar is longer only because of its three extra notebooks.

1. Line up the two purchases. Both have three pens, and the second has three more notebooks than the first.
2. The extra cost comes only from the three extra notebooks: 14.85 − 8.10 = $6.75.
3. (a) One notebook costs 6.75 ÷ 3 = $2.25.
4. Two notebooks cost 2 × 2.25 = $4.50, so the three pens cost 8.10 − 4.50 = $3.60.
5. (b) One pen costs 3.60 ÷ 3 = $1.20.

answer(a) $2.25; (b) $1.20

techniqueSubtracting in Columns · Dividing a Decimal · Money as Decimals

examsPSLE · GCSE Higher

Draw the full jar as the jar followed by two equal halves of rice. The second weighing is the jar followed by one half, so the two bars differ by one half of the rice.

1. Draw the full jar as the jar plus two equal halves of rice. The second weighing is the jar plus one half.
2. The difference between the two weighings is one half of the rice: 2.35 − 1.3 = 1.05 kg.
3. (a) All the rice has a mass of 2 × 1.05 = 2.1 kg.
4. (b) The empty jar has a mass of 2.35 − 2.1 = 0.25 kg. Check: 0.25 + 1.05 = 1.3.

answer(a) 2.1 kg; (b) 0.25 kg

techniqueSubtracting in Columns · Counting the Decimals

examsPSLE · GCSE Higher

Divide the length of the rope by the length of one piece. The whole-number part of the answer is the number of pieces, and the rope those pieces do not use is the leftover.

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.
2. 75 × 12 = 900 and 75 × 13 = 975, which is more than 920. So only 12 complete pieces fit.
3. (a) The greatest number of complete pieces is 12.
4. The 12 pieces use 12 × 0.75 = 9 m of rope.
5. (b) The length left over is 9.2 − 9 = 0.2 m.

answer(a) 12 pieces; (b) 0.2 m

techniqueDividing by a Decimal · Counting the Decimals

examsGCSE Higher

Draw a number line in hundredths from 4.6 to 4.8 and mark the numbers that are nearer to 4.7 than to 4.6 or 4.8.

1. Mark 4.7 on a number line in hundredths, with 4.6 and 4.8 on either side.
2. A number rounds up to 4.7 when its hundredths digit is 5 or more. The smallest such number is 4.65.
3. (a) The smallest possible mass is 4.65 kg.
4. A number rounds down to 4.7 when its hundredths digit is 4 or less. The largest such number with 2 decimal places is 4.74.
5. (b) The largest possible mass is 4.74 kg, because 4.75 rounds up to 4.8.

answer(a) 4.65 kg; (b) 4.74 kg

techniqueRounding Decimals to One and Two Decimal Places · Comparing Decimals

examsPSLE · O-Level · GCSE Higher

Two packs of different sizes cannot be compared until both prices are for the same mass. Find the price of 1 kg for each, then the cost of 6 kg made from each kind of pack.

1. Find the price of 1 kg in Pack A: 4.20 ÷ 0.75 = 420 ÷ 75 = $5.60.
2. Find the price of 1 kg in Pack B: 6.60 ÷ 1.2 = 66 ÷ 12 = $5.50.
3. (a) Pack A costs $5.60 per kilogram and Pack B costs $5.50 per kilogram, so 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.
5. (b) She saves 33.60 − 33 = $0.60.

answer(a) Pack A: $5.60 per kg, Pack B: $5.50 per kg; (b) $0.60

techniqueDividing by a Decimal · Counting the Decimals

examsGCSE Higher

Draw the posts as marks on a line. The distance is made of the gaps between the marks, and there is always one gap fewer than there are posts.

1. Count the gaps, not the posts. Twelve posts in a line have 12 − 1 = 11 gaps between them.
2. The 11 equal gaps make 46.2 m, so one gap is 46.2 ÷ 11 = 4.2 m.
3. (a) Neighboring lamp posts are 4.2 m apart.
4. From the 3rd post to the 9th there are 9 − 3 = 6 gaps.
5. (b) The distance is 6 × 4.2 = 25.2 m.

answer(a) 4.2 m; (b) 25.2 m

techniqueDividing a Decimal · Counting the Decimals

examsPSLE · GCSE Higher

Suppose every coin is a 20-cent coin and compare the supposed total with the real one. Each 50-cent coin that was counted as a 20-cent coin explains $0.30 of the shortfall.

1. Suppose all 40 coins were 20-cent coins. Their value would be 40 × 0.20 = $8.
2. The real total is $13.10, which is 13.10 − 8 = $5.10 more than the supposed total.
3. Changing one 20-cent coin into a 50-cent coin adds 0.50 − 0.20 = $0.30.
4. (a) The number of 50-cent coins is 5.10 ÷ 0.30 = 17.
5. (b) There are 40 − 17 = 23 20-cent coins, worth 23 × 0.20 = $4.60. Check: 17 × 0.50 + 4.60 = $13.10.

answer(a) 17 coins; (b) $4.60

techniqueCounting the Decimals · Dividing by a Decimal

examsPSLE · GCSE Higher

Start from the money left at the end and undo each step in reverse order: first the book, then the lunch.

1. Work backwards from the $6.40 he had left. The book cost half of his money, so the $6.40 is the other half.
2. (a) Just before he bought the book he had 2 × 6.40 = $12.80.
3. Before that he spent $3.75 on lunch, so add it back: 12.80 + 3.75 = $16.55.
4. (b) Ali had $16.55 at first. Check: 16.55 − 3.75 = 12.80, and half of 12.80 is 6.40.

answer(a) $12.80; (b) $16.55

techniqueMultiplying and Dividing Undo Each Other · Adding in Columns

examsPSLE · GCSE Higher

Write the four times in a place-value table with the decimal points lined up, and compare them one place at a time, starting from the left.

1. Line up the decimal points in a place-value table. Write zeros in the empty places so that every time has three decimal places: 47.407, 47.470, 47.400 and 47.074.
2. Every time has 47 whole seconds, so compare the tenths. Nora's time has 0 tenths and the other three have 4 tenths, so Nora's time is the shortest.
3. Compare the hundredths of the other three times. Lea's time has 7 hundredths and the other two have 0 hundredths, so Lea's time is the longest.
4. Compare the thousandths of Mia's and Kim's times. 0 thousandths is less than 7 thousandths, so 47.400 < 47.407. (a) From first place to last place: Nora, Mia, Kim, Lea.
5. (b) The winner beat the last slider by 47.470 − 47.074 = 0.396 s. Check: 47.074 + 0.396 = 47.470.

answer(a) Nora, Mia, Kim, Lea; (b) 0.396 s

techniqueThousandths · Comparing Decimals · Subtracting in Columns

examsPSLE · GCSE Higher

Draw the 20 kg limit as one bar. The packed suitcase fills part of it, and the three boxes must share the part that is left.

1. Draw the 20 kg limit as one bar. The suitcase and everything in it fill part of the bar, and the rest is the room left for the boxes.
2. Add the four masses in columns with the decimal points lined up. Write zeros so that each mass has three decimal places: 3.600 + 6.850 + 1.200 + 4.375 = 16.025.
3. (a) The packed suitcase has a mass of 16.025 kg.
4. The room left under the limit is 20 − 16.025 = 3.975 kg.
5. (b) The three boxes share that room equally, so each box can have a mass of at most 3.975 ÷ 3 = 1.325 kg. Check: 16.025 + 3 × 1.325 = 20.

answer(a) 16.025 kg; (b) 1.325 kg

techniqueLining Up the Dot · Adding in Columns · Dividing a Decimal

examsPSLE

Read each mark as a tenth and some hundredths, then count the hundredths along the scale from one mark up to the other.

1. The ten equal parts between two long lines split one tenth of a meter into ten, so each short line marks one hundredth of a meter, 0.01 m.
2. January's mark is 7 hundredths above 1.2: 1.2 + 0.07 = 1.27 m. July's mark is 5 hundredths above 1.3: 1.3 + 0.05 = 1.35 m.
3. (a) Mei was 1.27 m tall in January and 1.35 m tall in July.
4. Count up the scale from 1.27: 3 hundredths reach 1.3, and 5 more reach 1.35. That is 3 + 5 = 8 hundredths of a meter.
5. (b) One hundredth of a meter is 1 cm, so Mei grew 8 cm. Check: 1.35 − 1.27 = 0.08 m, which is 8 cm.

answer(a) January 1.27 m, July 1.35 m; (b) 8 cm

techniqueHundredths · Decimals · Subtracting in Columns

examsPSLE · GCSE Higher

Change the fraction to a decimal so that it can be multiplied. Then draw the flour needed against whole bags of flour.

1. Three quarters is 0.75, because 34 = 75100. So one batch needs 0.75 kg of flour.
2. Five batches need 5 × 0.75 = 3.75 kg of flour.
3. (a) She needs 3.75 kg of flour.
4. One bag holds only 2.5 kg, which is less than 3.75 kg. Two bags hold 2 × 2.5 = 5 kg, which is enough.
5. (b) She must buy 2 bags, and 5 − 3.75 = 1.25 kg of flour is left over. Check: 3.75 + 1.25 = 5.

answer(a) 3.75 kg; (b) 2 bags, with 1.25 kg left over

technique0.5 Is a Half · Counting the Decimals · Decimals

examsPSLE · SAT · GCSE Higher