Ratio

Percent, ratio and scale, grade 6 Lesson 7 of 11Learning

Ratio

The ratio of two quantities

A ratio compares two quantities by their quotient: the first quantity divided by the second. A class has 15 boys and 20 girls, so the ratio of the number of boys to the number of girls is $\displaystyle 15:20$ . Read it “15 to 20”.

A ratio is a quotient, so it can also be written as a fraction:

\displaystyle a:b

\displaystyle =

\displaystyle \frac{a}{b}

A ratio as a strip

This strip has 3 blue squares and 4 yellow squares:

The ratio of blue to yellow is $\displaystyle 3:4$ – for every 3 blue squares there are 4 yellow ones.

The quantities in a ratio

\displaystyle 15:20

and

\displaystyle 20:15

are two different ratios:

\displaystyle \frac{15}{20}

is less than 1, and

\displaystyle \frac{20}{15}

is more than 1.

A ratio needs its two quantities named. In the same class:

the ratio of boys to girls: $\displaystyle 15:20$ ;
the ratio of girls to boys: $\displaystyle 20:15$ ;
the ratio of boys to all the students: $\displaystyle 15:35$ , since the class has $\displaystyle {15+20}={35}$ students.

The first number in a ratio belongs to the quantity named first.

Reducing a ratio

A ratio is reduced like a fraction: divide both numbers by the same number. The quotient does not change, so the ratio does not change either. In a strip of 6 blue squares and 8 yellow ones the ratio is $\displaystyle {6:8}={3:4}$ – both numbers divided by 2. Here too, for every 3 blue squares there are 4 yellow ones.

Exercise 1

Reduce the ratios from the class of 15 boys and 20 girls.

Boys to girls

/ – 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

Boys to all the students

/ – 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

Girls to boys

/ – 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

A bowl holds 8 apples and 12 pears.
Write the ratio of pears to all the fruit in the bowl, and reduce it.

Exercise 3

The strip has 2 blue squares, and the rest are empty:

What is the ratio of blue squares to empty squares?
And the ratio of empty squares to all the squares?
Write both ratios reduced.

Formulas: percent, ratio and scale, grade 6

Open as a page

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
$\displaystyle b$ $\displaystyle =$ $\displaystyle a\times n$