Line up the points, fill the gap
To work out 5.3 − 2.18, write 2.18 under 5.3 with the decimal points lined up. 2.18 reaches into the hundredths, but 5.3 stops at the tenths, so the top number has an empty hundredths place.
5.3 has no hundredths, so write a 0 there: 5.30 is the same number as 5.3, because 3 tenths is 30 hundredths. Now both numbers have a hundredths digit.
The points are lined up. 5.3 has nothing in the hundredths column, above the 8 of 2.18.
The 0 fills the hundredths place of 5.30.
Borrow a tenth
Start with the hundredths: 0 − 8. There are not enough hundredths to take 8 away, so borrow from the tenths column. One tenth is ten hundredths.
Cross out the 3 tenths and write 2 above them, because one of them has been borrowed. The hundredths now hold 10. This is the same borrowing as in whole numbers, where one ten is exchanged for ten ones.
One tenth is exchanged for ten hundredths: the tenths drop from 3 to 2, and the hundredths hold 10.
Subtract each column
Now each column subtracts. The hundredths: 10 − 8 = 2. The tenths: 2 − 1 = 1. The ones: 5 − 2 = 3. Bring the point straight down into the answer: 5.3 − 2.18 = 3.12.
Adding checks it: 3.12 + 2.18 = 5.30, the number at the top.
Each column subtracts, and the point comes straight down: 3.12.
Two slips
Forgetting that the tenths went down from 3 to 2 gives 3.22, which is a tenth too much. The borrowed tenth has to be taken off the tenths column.
Taking 0 from 8 instead of 8 from 0 gives 3.28. The top number is always the one being taken from, so borrow when its digit is too small.
Worked example: A Container Weighed Full and Half Full
Question A jar filled completely with rice has a mass of 2.35 kg. After half of the rice is used, the jar and the remaining rice have a mass of 1.3 kg. (a) What was the mass of the rice when the jar was full? (b) What is the mass of the empty jar?
1.Draw the full jar as the jar plus two equal halves of rice. The second weighing is the jar plus one half.
The small block is the jar. The rice is two equal halves, and the second weighing has one of them. 2.The difference between the two weighings is one half of the rice: 2.35 − 1.3 = 1.05 kg.
The two weighings differ by one half of the rice: 1.05 kg. 3.(a) All the rice has a mass of 2 × 1.05 = 2.1 kg.
(a) All the rice is 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.
(b) The jar alone is 2.35 − 2.1 = 0.25 kg.
Answer: (a) 2.1 kg; (b) 0.25 kg
Common mistakes
- Halving 2.35 kg. Only the rice is halved. The mass of the jar is the same in both weighings.
- Writing 2.35 − 1.3 = 2.22 by subtracting 13 hundredths. 1.3 is the same as 1.30, so line up the decimal points before subtracting.
Worked example: Luge Times Measured to the Thousandth of a Second
Question In one run of a luge race, each slider is timed to the thousandth of a second. A reporter copied the four times into her notebook without their zeros at the end: Kim 47.407 s, Lea 47.47 s, Mia 47.4 s and Nora 47.074 s. The shortest time wins. (a) List the four sliders from first place to last place. (b) By how many seconds did the winner beat the slider who came last?
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.
Line up the points, and write zeros so that every time has three decimal places. 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.
The tenths: Nora has 0 and the others have 4, so Nora is first. 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.
The hundredths: Lea has 7 and the other two have 0, so Lea is last. 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.
(a) The thousandths: 47.400 < 47.407, so Mia is second and Kim third. 5.(b) The winner beat the last slider by 47.470 − 47.074 = 0.396 s. Check: 47.074 + 0.396 = 47.470.
(b) The winner beat the last slider by 47.470 − 47.074 = 0.396 s.
Answer: (a) Nora, Mia, Kim, Lea; (b) 0.396 s
Common mistakes
- Deciding that 47.407 is longer than 47.47 because 407 is more than 47. Compare place by place instead: both have 4 tenths, and 7 hundredths is more than 0 hundredths, so 47.47 is the longer time.
- Lining up the last digits instead of the decimal points when subtracting, so that the hundredths of 47.47 are taken from the thousandths of 47.074. Write 47.47 as 47.470 first.