Three ways of writing a part
Three shops sell the same jacket, and each one offers a discount. The first takes off the price, the second takes 0.3 off, and the third takes 27% off. Which discount is the biggest?
The numbers as they are written do not answer it. 27 is the biggest number on show, but 27% is not the biggest discount. is a fraction, 0.3 is a decimal and 27% is a percent, and a fraction, a decimal and a percent cannot be compared digit by digit. First write all three in the same form.
Write them all as percents
A percent is a number of hundredths, so turn each discount into hundredths.
For , multiply the numerator and the denominator by 25: , which is 25%.
0.3 is three tenths. Each tenth is ten hundredths, so three tenths are thirty hundredths: %.
27% is already a percent, so it stays as it is.
A quarter of the hundred square is 25 small squares, so is 25%.
Three tenths of the square are 3 full rows of 10, which is 30 small squares: 0.3 is 30%.
27% is 27 small squares out of 100: 2 full rows and 7 more.
Line them up
Now all three are hundredths of the same whole, and they can be put in order: 25 is less than 27, and 27 is less than 30. So 25% < 27% < 30%.
The biggest discount is 30%, which is the 0.3 off. The smallest is 25%, which is the off. Give the answer in the form the question used: the shop that takes 0.3 off gives the biggest discount.
As percents, the three discounts sit at 25, 27 and 30 on one line, and 30, the 0.3 off, is furthest along.
Any one form will do
Percents are not the only choice. Written as decimals, the three discounts are 0.25, 0.3 and 0.27. Give 0.3 a zero at the end, 0.30, so that all three have two decimal places. Then 0.25 < 0.27 < 0.30, the same order as before.
Two slips are common here. One is to think 0.25 is more than 0.3 because it has more digits; 0.3 is 30 hundredths, and that is more than 25 hundredths. The other is to read as big because its 4 is bigger than the 3 in 0.3. The 4 is the denominator: it says the whole is cut into 4 equal pieces, and is just one of them, 25 hundredths of the whole.
Worked example: Cycling Shares Given as a Fraction, a Decimal and a Percentage
Question Three schools recorded how many of their students cycle to school. At Northfield, which has 600 students, 720 of the students cycle. At Eastbrook, which has 750 students, the share of students who cycle is 0.32. At Westgate, which has 400 students, 36% of the students cycle. (a) Rank the three schools by the share of their students who cycle, largest share first. (b) Which school has the greatest number of students who cycle, and how many more cyclists does it have than the school with the largest share?
1.Northfield: 720 = 7 × 520 × 5 = 35100, so 35 parts out of 100 are shaded.
Northfield: 720 = 35100, so 35 parts of 100 are shaded. 2.Eastbrook: 0.32 means 32 hundredths, so 32 parts out of 100 are shaded.
Eastbrook: 0.32 is 32 hundredths, so 32 parts of 100. 3.Westgate: 36% means 36 out of 100, so 36 parts out of 100 are shaded.
Westgate: 36% is 36 parts of 100. 4.(a) Every bar has 100 parts, so the shaded parts can be compared directly. The order is Westgate (36%), Northfield (35%), Eastbrook (32%).
(a) All three are out of 100 parts: Westgate 36, Northfield 35, Eastbrook 32. 5.Now count the cyclists. One part is 1% of a school. At Northfield, 1% of 600 is 6 students, so 35% is 35 × 6 = 210 students.
Northfield: 1% of 600 is 6, so 35% is 210 students. 6.At Eastbrook, 1% of 750 is 7.5 students, so 32% is 32 × 7.5 = 240 students.
Eastbrook: 1% of 750 is 7.5, so 32% is 240 students. 7.At Westgate, 1% of 400 is 4 students, so 36% is 36 × 4 = 144 students.
Westgate: 1% of 400 is 4, so 36% is 144 students. 8.(b) Eastbrook has the most cyclists: 240 − 144 = 96 more than Westgate. Eastbrook's share is the smallest, but it is a share of the most students. Check: 144400 = 0.36 and 240750 = 0.32.
(b) Eastbrook has the most cyclists: 240 − 144 = 96 more than Westgate.
Answer: (a) Westgate 36%, Northfield 35%, Eastbrook 32%; (b) Eastbrook, 96 more
Common mistakes
- Comparing the numbers as they are written, so that 720 looks smaller than 0.32 or 36%. The three shares must be written in the same form, such as out of 100, before they can be compared.
- Taking the school with the largest share to have the most cyclists. Each share is out of that school's own number of students, so a smaller share of a larger school can be more students.