Dividing in a ratio

Percent, ratio and scale, grade 6 Lesson 9 of 11Learning

Dividing in a ratio

Dividing in rounds

Noa and Omer share 30 apples in the ratio $\displaystyle 2:3$ .
How many apples does each of them get?

The ratio $\displaystyle 2:3$ says: for every 2 apples Noa gets, Omer gets 3. We share in rounds:

in each round Noa gets 2 apples and Omer 3 – 5 apples together;
30 apples are enough for $\displaystyle {30\div5}={6}$ rounds;
after 6 rounds Noa has $\displaystyle {6\times2}={12}$ apples, and Omer $\displaystyle {6\times3}={18}$ .

The short way: equal parts

Think of the whole quantity as a strip of $\displaystyle {2+3}={5}$ equal parts – 2 parts for Noa and 3 for Omer:

The whole strip is 30 apples, so each part is $\displaystyle {30\div5}={6}$ apples.
Noa has 2 parts: $\displaystyle {2\times6}={12}$ apples.
Omer has 3 parts: $\displaystyle {3\times6}={18}$ apples.

Each part holds exactly as many apples as there were rounds: every round gives each part one apple.

Checking the division

At the end, check two things:

the whole quantity was shared: $\displaystyle {12+18}={30}$ ;
the ratio was kept: $\displaystyle {12:18}={2:3}$ .

Exercise 1

Two brothers share 35 sweets in the ratio $\displaystyle 3:4$ .
How many sweets does each of them get?

3 parts

/ – 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

4 parts

/ – 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

Exercise 2

Yael and Ron worked together in a neighbour’s garden and got 48 ₪.
Yael worked one hour and Ron 3 hours, so they share the money in the ratio $\displaystyle 1:3$ .
How much money does each of them get?

Yael

/ – 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

Ron

/ – 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

Exercise 3

A class has 28 students.
The ratio of boys to girls in the class is $\displaystyle 3:4$ .
Draw a strip of the ratio, and find how many girls are in the class.

/ – 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

Formulas: percent, ratio and scale, grade 6

A percent is a hundredth

$\displaystyle a$ percent are $\displaystyle a$ hundredths
$\displaystyle a\%$ $\displaystyle =$ $\displaystyle \frac{a}{100}$
The whole
$\displaystyle 100\%$ $\displaystyle =$ $\displaystyle 1$
A half
$\displaystyle 50\%$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
A quarter
$\displaystyle 25\%$ $\displaystyle =$ $\displaystyle \frac{1}{4}$
Three quarters
$\displaystyle 75\%$ $\displaystyle =$ $\displaystyle \frac{3}{4}$
A tenth
$\displaystyle 10\%$ $\displaystyle =$ $\displaystyle \frac{1}{10}$
A fifth
$\displaystyle 20\%$ $\displaystyle =$ $\displaystyle \frac{1}{5}$
An eighth
$\displaystyle 12\frac{1}{2}\%$ $\displaystyle =$ $\displaystyle \frac{1}{8}$

The value of a percent

$\displaystyle a\%$ of the whole quantity $\displaystyle b$
$\displaystyle a\%\times b$ $\displaystyle =$ $\displaystyle \frac{a}{100}\times b$
The percent and the quantity can change places: 18% of 50 is 50% of 18
$\displaystyle a\%\times b$ $\displaystyle =$ $\displaystyle b\%\times a$

Finding the percent

What percent $\displaystyle a$ is of $\displaystyle b$ – reduce or expand to the denominator 100; the numerator $\displaystyle p$ is the number of percent
$\displaystyle \frac{a}{b}=\frac{p}{100}=p\%$

Ratio

A ratio is a quotient – it can be written as a fraction
$\displaystyle a:b$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
Equal ratios – both numbers multiplied or divided by the same number $\displaystyle k$
$\displaystyle a:b$ $\displaystyle =$ $\displaystyle (a\times k):(b\times k)$ $\displaystyle k\neq0$
A quantity $\displaystyle c$ shared in the ratio $\displaystyle a:b$ – the first share ( $\displaystyle a$ parts out of $\displaystyle a+b$ )
$\displaystyle \frac{c}{a+b}\times a$
A quantity $\displaystyle c$ shared in the ratio $\displaystyle a:b$ – the second share ( $\displaystyle b$ parts out of $\displaystyle a+b$ )
$\displaystyle \frac{c}{a+b}\times b$

Scale

At the scale $\displaystyle 1:n$ – $\displaystyle a$ the length in the drawing, $\displaystyle b$ the real length, in the same unit
$\displaystyle b$ $\displaystyle =$ $\displaystyle a\times n$