Which score is better?
Sam scored 15 out of 20 on a spelling test and 16 out of 25 on a math test. 16 marks is more than 15, but the math test had more marks to win. The two scores are out of different totals, so the numbers 15 and 16 cannot be compared as they stand.
Each bar is one whole test. The first is cut into 20 parts with 15 colored, and the second into 25 parts with 16 colored.
Make both out of 100
Percent means out of 100, so write each score out of 100. 20 × 5 = 100, so cut each of the 20 parts into 5 smaller pieces. The bar now has 100 pieces, and the 15 colored parts have become 15 × 5 = 75 pieces. 15 out of 20 is 75 out of 100, which is 75%.
Multiplying the score and the total by the same number makes an equivalent fraction: . The colored part of the bar does not change; only the pieces get smaller.
Each of the 20 parts is cut into 5, making 100 pieces, and 75 of them are colored.
Do the same to the math score. 25 × 4 = 100, so cut each of the 25 parts into 4. The 16 colored parts become 16 × 4 = 64 pieces: , which is 64%.
Each of the 25 parts is cut into 4, making 100 pieces, and 64 of them are colored.
Now compare
Both scores are out of 100 now, so they can be compared directly: 75% is more than 64%. The spelling score, 15 out of 20, is the better one, even though 16 marks is more than 15.
On a line from 0% to 100%, 75% lies to the right of 64%.
When the total does not go into 100
Some totals do not go into 100 a whole number of times. 40 is one of them, so for 26 out of 40, simplify first. Halve both numbers: . Now 20 goes into 100, so multiply both by 5: , which is 65%.
That score can now be ranked against 18 out of 25. , which is 72%, so 18 out of 25 is the better score.
Compare the shares, not the marks
The usual mistake is to compare the marks as they are written: 16 is more than 15, so the math score looks better. But each mark on the 20-mark test is 5% of that test, and each mark on the 25-mark test is only 4% of it. Write both scores out of 100 first, then compare.
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.