Sec 1 · G3 (Express)

Ratio, rate and speed

Practise ratio, rate and speed with step-by-step help. Free account.

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}

📖 Worked example

The ratio of the number of red marbles to the number of blue marbles is $3 : 2$ . After 10 more blue marbles are put into the bag, the ratio becomes $1 : 1$ . Find the number of blue marbles now.

Step 1 of 4

The number of red marbles does not change: make its units the same in both ratios (LCM)

\text{LCM of } 3 \text{ and } 1 = 3

📖 Worked example

Of the fruits in a basket, $\dfrac{1}{3}$ are mangoes. The rest are apples and oranges in the ratio $1 : 2$ . There are 8 oranges. Find the number of mangoes.

Step 1 of 4

The rest is what is left of the whole: 1 − 1/3

1 - \dfrac{1}{3} = \dfrac{2}{3}

📖 Worked example

Siti, Devi and Kumar share $840. The ratio of Siti's share to the total of Devi's and Kumar's shares is $2 : 3$ . Devi receives $90 less than Kumar. Find the amount Devi receives.

Step 1 of 2

Devi and Kumar together have 3 of the 5 units (in dollars)

\dfrac{3}{5} \times 840 = 504

📖 Worked example

The aspect ratio of a rectangle is width : height. A video with an aspect ratio of $4 : 3$ is shown on a screen with an aspect ratio of $16 : 9$ . It is made as large as possible without being cut off, so it fills the full height of the screen, as shown in the diagram. Find the fraction of the screen that is not covered.

Step 1 of 3

Both have the same height: make the heights the same number of units (LCM of the second terms)

\text{picture } 4 : 3 = 12 : 9

\text{screen } 16 : 9 = 16 : 9

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

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}

📖 Worked example

A train 150 m long travels at a constant speed of 54 km/h. It takes 30 seconds to pass completely over a bridge, from the moment its front reaches the bridge to the moment its back leaves it. Find the length of the bridge, in metres.

Step 1 of 3

Change the speed to metres per second (÷ 3.6)

54 \text{ km/h} = \dfrac{54}{3.6}

\phantom{54 \text{ km/h}} = 15 \text{ m/s}

📖 Worked example

A bicycle travels at a constant speed of 30 km/h. Each of its wheels has a diameter of 70 cm and turns without slipping. Find the number of complete turns (revolutions) each wheel makes in one minute, correct to the nearest whole number.

Step 1 of 3

The distance travelled in one minute, in cm (1 km = 100 000 cm, 1 h = 60 min)

\dfrac{30 \times 100\,000}{60} = 50000 \text{ cm}

Practise this →

4. Distance–time graphs

On the graph	Means	Example
gradient $\left(\dfrac{\text{rise}}{\text{run}}\right)$	speed	$12$ km in $30$ min: $12 \div \dfrac{30}{60} = 24$ km/h
flat stage	stopped	from $30$ to $50$ min: a $20$ -minute stop
steeper	faster	$24$ km/h is steeper than $18$ km/h
going down	travelling back towards the start	back at $0$ : home again
average speed	$\dfrac{\text{total distance}}{\text{total time}}$ , stops included	$24 \div \dfrac{90}{60} = 16$ km/h

\displaystyle \text{speed} = \text{gradient}

📖 Worked example

The distance–time graph shows Ravi's cycle ride from home. Find Ravi's speed from A to B, in km/h.

Step 1 of 3

Read the distances at A and B: the distance travelled (the rise)

\text{Distance} = 16 - 8

\phantom{\text{Distance}} = 8 \text{ km}

📖 Worked example

The distance–time graph shows Nur's drive from Town P. Find Nur's average speed for the whole journey, in km/h.

Step 1 of 3

Total distance travelled

The journey ends

110

km from Town P.

Practise this →

5. Utilities and bills

Bill	Charged for	How
electricity	kWh used	current reading $-$ previous reading, $\times$ the rate
water	m³ used, in tiers	the first part at the lower rate, the rest at the higher rate
mobile phone	minutes over the free time	monthly fee $+$ extra minutes $\times$ the rate

Electricity: units used

\displaystyle 24805 - 24315

e.g.

\displaystyle 24805 - 24315

\displaystyle = 490

at $27.5$ cents per kWh: $490 \times 27.5$ is $13\,475$ cents, which is $134.75

Water: 45 m³, the first 30 m³ at $1.20, the rest at $1.60

\displaystyle 30 \times 1.20 + 15 \times 1.60

e.g.

\displaystyle 30 \times 1.20 + 15 \times 1.60

\displaystyle = 60

📖 Worked example

The table shows a household's electricity meter readings for a month and the rate charged. Find the cost of the electricity used, in dollars.

Previous meter reading	Current meter reading	Rate
12460 kWh	12828 kWh	30 cents per kWh

Step 1 of 3

Units used = current reading − previous reading

12828 - 12460 = 368 \text{ kWh}

📖 Worked example

Mei Ling's mobile phone plan is shown in the table. In March, Mei Ling made 380 minutes of calls. Find the amount of Mei Ling's bill for March.

Monthly fee	Free talk time	Extra calls
$20	300 min	10 cents per min

Step 1 of 3

Extra minutes = minutes used − free minutes

380 - 300 = 80 \text{ min}

Practise this →

6. Currency exchange

A conversion graph of yen against Singapore dollars, a straight line through the origin: S$150 reads across to ¥16 200, and ¥10 800 reads down to S$100.

Rate	S$ to the foreign currency	Back to S$
S$1 buys ¥108	multiply by $108$	divide by $108$
1 ringgit costs S$0.30	divide by $0.30$	multiply by $0.30$

S$200 into yen

\displaystyle 200 \times 108

e.g.

\displaystyle 200 \times 108

\displaystyle = 21600

¥ $21\,600$

¥16 956 back into Singapore dollars

\displaystyle \dfrac{16956}{108}

e.g.

\displaystyle \dfrac{16956}{108}

\displaystyle = 157

S$157

📖 Worked example

The table shows a money changer's exchange rates. Devi changes S$800 into Malaysian ringgit. Find the amount, in Malaysian ringgit, that Devi receives. Give your answer correct to 2 decimal places where necessary.

Currency	S$1 buys
Malaysian ringgit (RM)	3.4
Japanese yen (¥)	108
Australian dollars (A$)	1.2

Step 1 of 1

Use the exchange rate: multiply or divide?

800 \times 3.4

= 2720

Devi receives RM 2720.

📖 Worked example

Siti wants to change S$1000 into Malaysian ringgit. The rates at two money changers are shown in the table. Which money changer gives Siti more Malaysian ringgit?

Money changer	Rate
A	S$1 = RM 3.4
B	S$100 = RM 346.8

Step 1 of 3

Money changer A: the amount of Malaysian ringgit received

\text{A: } 1000 \times 3.4 = 3400

📖 Worked example

On a certain day, S$1 $=$ RM 3.4 and S$1 $=$ US$0.75. Devi changes RM 100 into US dollars, through Singapore dollars. Find the amount, in US dollars, that Devi receives, correct to 2 decimal places.

Step 1 of 2

RM into Singapore dollars: divide by 3.4 (keep the full value)

\dfrac{100}{3.4}

= 29.4117\ldots

📖 Worked example

Before a holiday, Ben changes S$500 into Malaysian ringgit when S$1 $=$ RM 3.45. Ben spends RM 1,210 on the holiday. After the holiday, Ben changes the rest back into Singapore dollars when S$1 $=$ RM 3.56. Find the amount, in Singapore dollars, that Ben receives, correct to the nearest cent.

Step 1 of 3

Into Malaysian ringgit at the first rate: multiply

500 \times 3.45 = 1725

📖 Worked example

A tablet costs S$189 in Singapore. The same tablet costs RM 614 overseas, where S$1 $=$ RM 3.42. Where is it cheaper? Show your working.

Step 1 of 2

Change the overseas price into Singapore dollars: divide by the rate

\dfrac{614}{3.42}

= 179.5322\ldots

\approx 179.53

(to the nearest cent)

📖 Worked example

A camera costs S$250 in Singapore. Overseas, the same camera costs A$270, which is 10% cheaper than the price in Singapore. Find the exchange rate, giving your answer in the form S$1 $=$ A$ $x$ .

Step 1 of 2

The overseas price is worth (100% − p%) of the Singapore price (in dollars)

\dfrac{90}{100} \times 250 = 225

Practise this →

Try a question: Ratio, rate and speed

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise ratio, rate and speed with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free

Practice questions by skill

Sec 1 · G3 (Express)

Ratio, rate and speed

Practise ratio, rate and speed with step-by-step help. Free account.

Start free Worksheet (PDF)

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}

📖 Worked example

The ratio of the number of red marbles to the number of blue marbles is $3 : 2$ . After 10 more blue marbles are put into the bag, the ratio becomes $1 : 1$ . Find the number of blue marbles now.

Step 1 of 4

The number of red marbles does not change: make its units the same in both ratios (LCM)

\text{LCM of } 3 \text{ and } 1 = 3

📖 Worked example

Of the fruits in a basket, $\dfrac{1}{3}$ are mangoes. The rest are apples and oranges in the ratio $1 : 2$ . There are 8 oranges. Find the number of mangoes.

Step 1 of 4

The rest is what is left of the whole: 1 − 1/3

1 - \dfrac{1}{3} = \dfrac{2}{3}

📖 Worked example

Siti, Devi and Kumar share $840. The ratio of Siti's share to the total of Devi's and Kumar's shares is $2 : 3$ . Devi receives $90 less than Kumar. Find the amount Devi receives.

Step 1 of 2

Devi and Kumar together have 3 of the 5 units (in dollars)

\dfrac{3}{5} \times 840 = 504

📖 Worked example

Step 1 of 3

Both have the same height: make the heights the same number of units (LCM of the second terms)

\text{picture } 4 : 3 = 12 : 9

\text{screen } 16 : 9 = 16 : 9

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}

📖 Worked example

Step 1 of 3

Change the speed to metres per second (÷ 3.6)

54 \text{ km/h} = \dfrac{54}{3.6}

\phantom{54 \text{ km/h}} = 15 \text{ m/s}

📖 Worked example

Step 1 of 3

The distance travelled in one minute, in cm (1 km = 100 000 cm, 1 h = 60 min)

\dfrac{30 \times 100\,000}{60} = 50000 \text{ cm}

Practise this →

4. Distance–time graphs

On the graph	Means	Example
gradient $\left(\dfrac{\text{rise}}{\text{run}}\right)$	speed	$12$ km in $30$ min: $12 \div \dfrac{30}{60} = 24$ km/h
flat stage	stopped	from $30$ to $50$ min: a $20$ -minute stop
steeper	faster	$24$ km/h is steeper than $18$ km/h
going down	travelling back towards the start	back at $0$ : home again
average speed	$\dfrac{\text{total distance}}{\text{total time}}$ , stops included	$24 \div \dfrac{90}{60} = 16$ km/h

\displaystyle \text{speed} = \text{gradient}

📖 Worked example

The distance–time graph shows Ravi's cycle ride from home. Find Ravi's speed from A to B, in km/h.

Step 1 of 3

Read the distances at A and B: the distance travelled (the rise)

\text{Distance} = 16 - 8

\phantom{\text{Distance}} = 8 \text{ km}

📖 Worked example

The distance–time graph shows Nur's drive from Town P. Find Nur's average speed for the whole journey, in km/h.

Step 1 of 3

Total distance travelled

The journey ends

110

km from Town P.

Practise this →

5. Utilities and bills

Bill	Charged for	How
electricity	kWh used	current reading $-$ previous reading, $\times$ the rate
water	m³ used, in tiers	the first part at the lower rate, the rest at the higher rate
mobile phone	minutes over the free time	monthly fee $+$ extra minutes $\times$ the rate

Electricity: units used

\displaystyle 24805 - 24315

e.g.

\displaystyle 24805 - 24315

\displaystyle = 490

at $27.5$ cents per kWh: $490 \times 27.5$ is $13\,475$ cents, which is $134.75

Water: 45 m³, the first 30 m³ at $1.20, the rest at $1.60

\displaystyle 30 \times 1.20 + 15 \times 1.60

e.g.

\displaystyle 30 \times 1.20 + 15 \times 1.60

\displaystyle = 60

📖 Worked example

The table shows a household's electricity meter readings for a month and the rate charged. Find the cost of the electricity used, in dollars.

Previous meter reading	Current meter reading	Rate
12460 kWh	12828 kWh	30 cents per kWh

Step 1 of 3

Units used = current reading − previous reading

12828 - 12460 = 368 \text{ kWh}

📖 Worked example

Mei Ling's mobile phone plan is shown in the table. In March, Mei Ling made 380 minutes of calls. Find the amount of Mei Ling's bill for March.

Monthly fee	Free talk time	Extra calls
$20	300 min	10 cents per min

Step 1 of 3

Extra minutes = minutes used − free minutes

380 - 300 = 80 \text{ min}

Practise this →

6. Currency exchange

Rate	S$ to the foreign currency	Back to S$
S$1 buys ¥108	multiply by $108$	divide by $108$
1 ringgit costs S$0.30	divide by $0.30$	multiply by $0.30$

S$200 into yen

\displaystyle 200 \times 108

e.g.

\displaystyle 200 \times 108

\displaystyle = 21600

¥ $21\,600$

¥16 956 back into Singapore dollars

\displaystyle \dfrac{16956}{108}

e.g.

\displaystyle \dfrac{16956}{108}

\displaystyle = 157

S$157

📖 Worked example

Currency	S$1 buys
Malaysian ringgit (RM)	3.4
Japanese yen (¥)	108
Australian dollars (A$)	1.2

Step 1 of 1

Use the exchange rate: multiply or divide?

800 \times 3.4

= 2720

Devi receives RM 2720.

📖 Worked example

Siti wants to change S$1000 into Malaysian ringgit. The rates at two money changers are shown in the table. Which money changer gives Siti more Malaysian ringgit?

Money changer	Rate
A	S$1 = RM 3.4
B	S$100 = RM 346.8

Step 1 of 3

Money changer A: the amount of Malaysian ringgit received

\text{A: } 1000 \times 3.4 = 3400

📖 Worked example

On a certain day, S$1 $=$ RM 3.4 and S$1 $=$ US$0.75. Devi changes RM 100 into US dollars, through Singapore dollars. Find the amount, in US dollars, that Devi receives, correct to 2 decimal places.

Step 1 of 2

RM into Singapore dollars: divide by 3.4 (keep the full value)

\dfrac{100}{3.4}

= 29.4117\ldots

📖 Worked example

Step 1 of 3

Into Malaysian ringgit at the first rate: multiply

500 \times 3.45 = 1725

📖 Worked example

A tablet costs S$189 in Singapore. The same tablet costs RM 614 overseas, where S$1 $=$ RM 3.42. Where is it cheaper? Show your working.

Step 1 of 2

Change the overseas price into Singapore dollars: divide by the rate

\dfrac{614}{3.42}

= 179.5322\ldots

\approx 179.53

(to the nearest cent)

📖 Worked example

A camera costs S$250 in Singapore. Overseas, the same camera costs A$270, which is 10% cheaper than the price in Singapore. Find the exchange rate, giving your answer in the form S$1 $=$ A$ $x$ .

Step 1 of 2

The overseas price is worth (100% − p%) of the Singapore price (in dollars)

\dfrac{90}{100} \times 250 = 225

Practise this →

Try a question: Ratio, rate and speed

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise ratio, rate and speed with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free