Fraction and Ratio

Discover the connection between fractions and ratios

1. Converting Fractions to Ratios

The Key Insight

When we say "A is $\frac{a}{b}$ of B", it means:

A : B = a : b

Example 1: Leila's and Siti's Stickers

The number of Leila's stickers is $\frac{1}{2}$ of Siti's stickers.

Leila has 1 part, Siti has 2 parts

$\frac{1}{2}$ of Siti's → Ratio 1 : 2

Leila : Siti = 1 : 2

We can also say: Siti has twice as many stickers as Leila!

Example 2: Mike's and Sam's Marbles

The number of Mike's marbles is $\frac{3}{5}$ of Sam's marbles.

Mike has 3 parts, Sam has 5 parts

$\frac{3}{5}$ of Sam's → Ratio 3 : 5

Mike : Sam = 3 : 5

What about the other way around?

Sam : Mike = 5 : 3

So the number of Sam's marbles is $\frac{5}{3}$ of the number of Mike's marbles!

2. Converting Ratios to Fractions

Two Types of Fractions from Ratios

If A : B = a : b, we can find:

Part-to-Part: A is $\frac{a}{b}$ of B
Part-to-Whole: A is $\frac{a}{a+b}$ of total

Example 3: Mass of Parcels

The ratio of the mass of Parcel A to the mass of Parcel B is 4 : 5.

Part-to-Part:

Mass of A = $\frac{4}{5}$ of mass of B

Mass of B = $\frac{5}{4}$ of mass of A

Part-to-Whole:

Mass of A = $\frac{4}{9}$ of total mass

Mass of B = $\frac{5}{9}$ of total mass

Example 4: Water in Jugs

The volume of water in Jug X is $\frac{11}{12}$ of the volume in Jug Y.

Jug X : Jug Y = 11 : 12

(i) Ratio of Jug Y to Jug X = 12 : 11

(ii) Jug X as a fraction of the total = $\frac{11}{23}$

(iii) Jug Y as a fraction of Jug X = $\frac{12}{11}$

(iv) Ratio of the total volume to the volume in Jug X = 23 : 11

3. The Deep Connection

A Fraction Gives a Ratio

When A is $\frac{a}{b}$ of B, the ratio A : B is a : b.

Take care what B is: if 3/5 of a class are girls, girls : whole class = 3 : 5, but girls : boys = 3 : 2.

Example 5: Ribbons

Lina has a pink, a red and a blue ribbon. Their lengths are in the table. (i) What fraction of the length of the red ribbon is the length of the pink ribbon? (ii) What is the ratio of the length of the red ribbon to the length of the blue ribbon? (iii) The ratio of the length of one ribbon to the length of the pink ribbon is 7 : 18. Which ribbon is it?

Colour	Pink	Red	Blue
Length	36 cm	42 cm	14 cm

Divide all by 2 to simplify

(i) Pink as a fraction of red = $\frac{36}{42} = \frac{6}{7}$

(ii) Red : Blue = 42 : 14 = 3 : 1

(iii) Blue : Pink = 14 : 36 = 7 : 18, so it is the blue ribbon.

Quick Reference

Fraction Statement	Ratio
A is $\frac{1}{2}$ of B	A : B = 1 : 2
A is $\frac{2}{3}$ of B	A : B = 2 : 3
A is $\frac{3}{4}$ of B	A : B = 3 : 4
A is $\frac{5}{8}$ of B	A : B = 5 : 8

Key Points to Remember

✓ "A is a/b of B" means A : B = a : b
✓ If A : B = a : b, then A is a/(a + b) of the total
✓ If 3/5 of a class are girls, girls : whole class = 3 : 5, but girls : boys = 3 : 2
✓ Part-to-Part fraction uses just the two parts
✓ Part-to-Whole fraction uses the part and the total