Scale

Percent, ratio and scale, grade 6 Lesson 10 of 11Learning

Scale

Scale: the ratio of a drawing to reality

A map or a plan of a room is a small drawing of a real place, in the same shape. So there is a fixed ratio between every length in the drawing and the real length it stands for. That ratio is the drawing’s scale.

A scale is written so that one of its numbers is 1.
The first number is the length in the drawing, the second the real length.
A scale of $\displaystyle 1:100$ : every 1 cm in the drawing stands for 100 cm in reality, that is 1 m.

A classroom’s plan

This is the plan of a classroom at a scale of $\displaystyle 1:100$ .
The plan is drawn on squared paper, and a square’s side is 1 cm.

The classroom is 8 cm long on the plan. Every centimetre of the plan is 100 cm in reality, so the real classroom is $\displaystyle {8\times100}={800}$ cm long, that is 8 m.

A map’s scale

\displaystyle 100{,}000

cm are

\displaystyle {100{,}000\div100}={1000}

m, that is 1 km.

Many maps have the scale $\displaystyle 1:100{,}000$ : every 1 cm on the map stands for $\displaystyle 100{,}000$ cm in reality – exactly 1 km.

A scale has no unit: it is a ratio of two lengths in the same unit. So it holds in any unit – 8 mm on the map stand for $\displaystyle {8\times100{,}000}={800{,}000}$ mm in reality too.

An enlarging scale

A drawing of something small, like an ant or a screw, is bigger than the real thing. At a scale of $\displaystyle 3:1$ every 1 cm in reality is 3 cm in the drawing – the drawing is 3 times bigger. Here too one of the numbers is 1, but this time it is the second one.

Area at a scale

A scale is a ratio of lengths. On a plan at a scale of $\displaystyle 1:100$ , a square of 1 cm by 1 cm stands for a square of 100 cm by 100 cm – that is 1 m by 1 m. The square’s area on the plan is 1 cm², and in reality $\displaystyle {100\times100}={10{,}000}$ cm² – not 100 cm².

Exercise 1

Write the scale so that one of its numbers is 1.

a

2 cm on the plan stand for 500 cm

/ – 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

b

5 cm on the map stand for

\displaystyle 1{,}000{,}000

cm

/ – 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

c

4 cm in the drawing stand for 2 cm

/ – 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

Go back to the classroom’s plan at a scale of $\displaystyle 1:100$ .
How wide is the real classroom, in metres?

/ – 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

On a map at a scale of $\displaystyle 1:50{,}000$ two villages are 6 cm apart.
What is the real distance between the villages, in kilometres?

/ – 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 4

A room is a rectangle 4 m long and 3 m wide.
Draw its plan at a scale of $\displaystyle 1:100$ .
A square’s side in the drawing is 1 cm.

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