One amount
Cut a bar into 4 equal pieces and color 3 of them. The colored part is three quarters of the bar, written .
The bar is cut into 4 equal pieces, and 3 of them are colored.
The same amount in hundredths
Now cut the same whole into 100 equal pieces instead, like a hundred square. Each quarter is 100 ÷ 4 = 25 of the small squares, so three quarters are 3 × 25 = 75 of them. .
75 hundredths is written as a decimal as 0.75: 7 tenths and 5 hundredths. So .
The hundred square has 75 of its 100 small squares colored: 75 hundredths, 0.75.
And as a percent
Percent means out of 100, so 75 out of 100 is 75%. The colored amount never changed: , 0.75 and 75% are three ways of writing the same amount.
Going from one to another
For a fraction whose denominator goes into 100, multiply the numerator and the denominator by the same number to reach 100. 20 × 5 = 100, so , which is 0.15 and 15%.
From a percent, go the other way. 40% is 40 out of 100, so it is , which simplifies to , and it is 0.4 as a decimal.
From a decimal, read it in hundredths. 0.3 is 3 tenths, and 3 tenths are 30 hundredths, so 0.3 = 30%, not 3%.
a percent is a hundredth, so n% of a quantity is that quantity × n/100
Make a quarter
The bar opens at 40%, which is . Drag it to a quarter of the bar: a quarter is 25 hundredths, so it reads 25%.
Fractions worth knowing
A few fractions come up so often that it is worth knowing all three names by heart. %. %, and %. %, and %.
A denominator that does not go into 100 needs a division. , which is 12.5%: twelve and a half hundredths.
3/4 = 0.75
Make one half
The bar opens at . Make one half: , and all fill half the bar, and a half is 50 out of 100, which is 0.5 or 50%.
Two slips
A decimal and a percent of the same amount are written differently. is 0.75, and it is 75%. Writing 0.75% would mean less than one hundredth, far less than three quarters.
And is not 34%. The fraction means 3 ÷ 4, which is 75 out of 100; it is not the digits 3 and 4 written side by side.
Worked example: Charity Fair Money Shared Out as a Fraction, a Decimal and a Percentage
Question A club raised $4800 at a charity fair. It gave 38 of the money to a children's hospital, 0.3 of the money to an animal shelter and 20% of the money to the town library, and kept the rest for next year's fair. (a) How much money did the club keep? (b) What fraction of the money that the club gave away went to the children's hospital?
1.Hospital: 38 = 3 × 58 × 5 = 1540, so the hospital gets 15 units.
The $4800 is 40 units. Hospital: 38 = 1540, so 15 units. 2.Shelter: 0.3 = 310 = 1240, so the shelter gets 12 units.
Shelter: 0.3 = 1240, so 12 units. 3.Library: 20% = 20100 = 15 = 840, so the library gets 8 units.
Library: 20% = 840, so 8 units. 4.Given away: 15 + 12 + 8 = 35 units, so the club kept 40 − 35 = 5 units.
Given away: 15 + 12 + 8 = 35 units, so 5 units are kept. 5.1 unit = 4800 ÷ 40 = $120. (a) The club kept 5 × 120 = $600.
1 unit is 4800 ÷ 40 = $120. (a) Kept: 5 × 120 = $600. 6.(b) The club gave away 35 units, and 15 of them went to the hospital: 1535 = 37. Check: $1800 + $1440 + $960 + $600 = $4800.
(b) The hospital has 15 of the 35 units given away: 1535 = 37.
Answer: (a) $600; (b) 37
Common mistakes
- Adding the numbers as they are written, 38 + 0.3 + 20, which mixes three forms. Each share must be written in the same form, such as fortieths or percentages, before the shares can be added.
- Answering (b) with 38. That is the hospital's share of all the money raised, but (b) asks for its share of the $4200 given away, which is a smaller whole.