Sec 1 · G1 (N(T))

Ratio, rate and speed

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1. Ratio

Rule	Example
divide by the HCF	$12 : 18 = 2 : 3$
same units first	$250\text{ g} : 1\text{ kg}$ is $250 : 1000$ , so $1 : 4$
fractions: multiply by the LCM of the denominators	$\dfrac{1}{2} : \dfrac{2}{3} = 3 : 4$
three quantities	$a : b = 2 : 3$ and $b : c = 4 : 5$ give $8 : 12 : 15$

Share 64 in the ratio 3 : 5: find 1 unit first

\displaystyle 3 + 5 = 8 \text{ units}

Bar model: A is 3 boxes and B is 5 boxes, each box 8, with a total of 64.

e.g.

\displaystyle \dfrac{64}{8}

\displaystyle = 8

A: $3 \times 8$ , which is $24$
B: $5 \times 8$ , which is $40$

📖 Worked example

Express the ratio $200$ g $:$ $3$ kg in its simplest form.

Step 1 of 2

Change both quantities to the same (smaller) unit first

3 \text{ kg} = 3 \times 1000 \text{ g}

\phantom{3 \text{ kg}} = 3000 \text{ g}

📖 Worked example

Siti and Devi share 96 stamps in the ratio $5 : 3$ . Find the number of stamps Devi gets.

Step 1 of 3

Count the total number of units

\text{Total} = 5 + 3

\phantom{\text{Total}} = 8 \text{ units}

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2. Rate

Rate	Example
per unit	$4.50 for 3 kg is $1.50 per kg
printing	$120$ pages in $8$ minutes is $15$ pages per minute

📖 Worked example

A printer prints 270 pages in 9 minutes. Find its printing rate, in pages per minute.

Step 1 of 1

Find the rate: multiply or divide?

\text{Rate} = \dfrac{270}{9}

= 30

The rate is

30

pages per minute.

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3. Speed

Find	Formula
speed	$\dfrac{\text{distance}}{\text{time}}$
distance	speed $\times$ time
time	$\dfrac{\text{distance}}{\text{speed}}$
average speed	$\dfrac{\text{total distance}}{\text{total time}}$

Kilometres per hour to metres per second: × 1000, then ÷ 3600

\displaystyle 72\text{ km per hour}

e.g.

\displaystyle \dfrac{72 \times 1000}{3600}

\displaystyle = 20

$20$ m/s

Average speed: 60 km in 1.5 h, then 40 km in 0.5 h

\displaystyle \dfrac{60 + 40}{1.5 + 0.5}

e.g.

\displaystyle \dfrac{60 + 40}{1.5 + 0.5}

\displaystyle = 50

$50$ km/h, not the mean of $40$ and $80$

📖 Worked example

A train travels 120 km at a constant speed of 50 km/h. Find the time taken. Give your answer in hours and minutes.

Step 1 of 2

Find the time in hours: multiply or divide?

\text{Time} = \dfrac{120}{50}

= 2.4

📖 Worked example

A car travels from A to B and then from B to C, as shown in the diagram. The journey from A to B takes 2 hours and the journey from B to C takes 1 hour. Find the average speed for the whole journey, in km/h.

Step 1 of 3

Total distance

\text{Total distance} = 90 + 54

\phantom{\text{Total distance}} = 144 \text{ km}

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1. Ratio

Rule	Example
divide by the HCF	$12 : 18 = 2 : 3$
same units first	$250\text{ g} : 1\text{ kg}$ is $250 : 1000$ , so $1 : 4$
fractions: multiply by the LCM of the denominators	$\dfrac{1}{2} : \dfrac{2}{3} = 3 : 4$
three quantities	$a : b = 2 : 3$ and $b : c = 4 : 5$ give $8 : 12 : 15$

Share 64 in the ratio 3 : 5: find 1 unit first

\displaystyle 3 + 5 = 8 \text{ units}

e.g.

\displaystyle \dfrac{64}{8}

\displaystyle = 8

A: $3 \times 8$ , which is $24$
B: $5 \times 8$ , which is $40$

📖 Worked example

Express the ratio $200$ g $:$ $3$ kg in its simplest form.

Step 1 of 2

Change both quantities to the same (smaller) unit first

3 \text{ kg} = 3 \times 1000 \text{ g}

\phantom{3 \text{ kg}} = 3000 \text{ g}

📖 Worked example

Siti and Devi share 96 stamps in the ratio $5 : 3$ . Find the number of stamps Devi gets.

Step 1 of 3

Count the total number of units

\text{Total} = 5 + 3

\phantom{\text{Total}} = 8 \text{ units}

Practise this →

2. Rate

Rate	Example
per unit	$4.50 for 3 kg is $1.50 per kg
printing	$120$ pages in $8$ minutes is $15$ pages per minute

📖 Worked example

A printer prints 270 pages in 9 minutes. Find its printing rate, in pages per minute.

Step 1 of 1

Find the rate: multiply or divide?

\text{Rate} = \dfrac{270}{9}

= 30

The rate is

30

pages per minute.

Practise this →

3. Speed

Find	Formula
speed	$\dfrac{\text{distance}}{\text{time}}$
distance	speed $\times$ time
time	$\dfrac{\text{distance}}{\text{speed}}$
average speed	$\dfrac{\text{total distance}}{\text{total time}}$

Kilometres per hour to metres per second: × 1000, then ÷ 3600

\displaystyle 72\text{ km per hour}

e.g.

\displaystyle \dfrac{72 \times 1000}{3600}

\displaystyle = 20

$20$ m/s

Average speed: 60 km in 1.5 h, then 40 km in 0.5 h

\displaystyle \dfrac{60 + 40}{1.5 + 0.5}

e.g.

\displaystyle \dfrac{60 + 40}{1.5 + 0.5}

\displaystyle = 50

$50$ km/h, not the mean of $40$ and $80$

📖 Worked example

A train travels 120 km at a constant speed of 50 km/h. Find the time taken. Give your answer in hours and minutes.

Step 1 of 2

Find the time in hours: multiply or divide?

\text{Time} = \dfrac{120}{50}

= 2.4

📖 Worked example

Step 1 of 3

Total distance

\text{Total distance} = 90 + 54

\phantom{\text{Total distance}} = 144 \text{ km}

Practise this →