A ratio of three quantities

Ratio, proportion and percent, grade 8 Lesson 1 of 12Learning

A ratio of three quantities

Recap: ratio

A ratio compares two quantities: a class has 15 boys and 20 girls, and the ratio of boys to girls is $\displaystyle 15:20$ .

A ratio is simplified and expanded like a fraction: $\displaystyle {15:20}={3:4}$ .
The order matters: the ratio of girls to boys is $\displaystyle 4:3$ .

A ratio of quantities in the same unit

1 m = 100 cm
1 km = 1000 m
1 kg = 1000 g
1 hour = 60 minutes

A ratio compares quantities of the same kind – a length with a length, a time with a time – so both must be in the same unit.
A table is 2 m long, and a drawer in it is 50 cm long.
$\displaystyle 2:50$ is not their ratio, because 2 is in metres and 50 in centimetres.

Write both lengths in centimetres: 2 m is 200 cm.
The ratio of the table’s length to the drawer’s: $\displaystyle {200:50}={4:1}$ – the table is 4 times as long.

Exercise 1

Write as a ratio in simplest form.

40 cm to 2 m

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

45 minutes to an hour

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

750 g to 2 kg

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A ratio of three quantities

A ratio can compare three quantities too.
A cake recipe has 2 cups of sugar, 3 cups of flour and 1 cup of milk.
The ratio of sugar to flour to milk is $\displaystyle 2:3:1$ .

It is read “two to three to one”.
Each number belongs to the quantity in its place: the first to the sugar, the second to the flour, the third to the milk.
As in a ratio of two quantities, the order matters.

Simplifying and expanding

A ratio of three quantities is simplified and expanded too: divide or multiply all three numbers by the same number.

$\displaystyle {12:18:30}={2:3:5}$ – divided by 6.
$\displaystyle {0.5:1:1.5}={1:2:3}$ – multiplied by 2.
A double cake: $\displaystyle {2:3:1}={4:6:2}$ – 4 cups of sugar, 6 cups of flour and 2 cups of milk.

Exercise 2

Write as a ratio in simplest form.

$\displaystyle 8:12:20$
$\displaystyle 15:45:30$
$\displaystyle 0.2:0.6:1$

Dividing in a ratio of three

Three friends share 60 ₪ in the ratio $\displaystyle 1:2:3$ .
How much money does each get?

The whole amount is $\displaystyle {1+2+3}={6}$ equal parts.
Each part is $\displaystyle {60:6}={10}$ ₪.
The first gets one part – 10 ₪, the second 2 parts – 20 ₪, and the third 3 parts – 30 ₪.
Check: $\displaystyle {10+20+30}={60}$ , and $\displaystyle {10:20:30}={1:2:3}$ .

Exercise 3

Three sisters share 90 ₪ in the ratio $\displaystyle 2:3:4$ .
How much money does each get?

First sister

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Second sister

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Third sister

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The angles of a triangle add up to

\displaystyle 180^\circ

The angles of a triangle are in the ratio $\displaystyle 1:2:3$ .
Find the angles.

Smallest angle

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Middle angle

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Largest angle

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: ratio, proportion and percent, grade 8

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Ratio and proportion

A ratio is also a fraction
$\displaystyle a:b$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
Cross-multiplying a proportion
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$
Quantities in the ratio $\displaystyle a:b$ – $\displaystyle k$ the size of one part
$\displaystyle ak:bk$ $\displaystyle =$ $\displaystyle a:b$
Directly proportional quantities – the ratio $\displaystyle k$ stays the same (the unit price, for one)
$\displaystyle \frac{y}{x}=k$

Scale

At the scale $\displaystyle 1:n$ – $\displaystyle a$ the length in the drawing, $\displaystyle b$ the real length, in the same unit
$\displaystyle b$ $\displaystyle =$ $\displaystyle a\cdot n$
An area at the scale $\displaystyle 1:n$ – it grows $\displaystyle n^2$ times, not $\displaystyle n$ times
$\displaystyle S_b$ $\displaystyle =$ $\displaystyle S_a\cdot n^2$

Percent and percent change

$\displaystyle p$ percent are $\displaystyle p$ hundredths
$\displaystyle p\%$ $\displaystyle =$ $\displaystyle \frac{p}{100}$
A rise of $\displaystyle p\%$ – $\displaystyle a$ the amount before the change, $\displaystyle b$ the amount after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%+p\%)\cdot a$
A fall of $\displaystyle p\%$ – $\displaystyle a$ the amount before the change, $\displaystyle b$ the amount after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%-p\%)\cdot a$
The percent change – a part of the amount before the change
$\displaystyle p\%=\frac{b-a}{a}$
Two changes one after the other – a rise of $\displaystyle p\%$ , then a rise of $\displaystyle q\%$ (for a fall: minus instead of plus)
$\displaystyle c$ $\displaystyle =$ $\displaystyle (100\%+p\%)\cdot(100\%+q\%)\cdot a$