Chapter 6 of 8

All 8 chapters
  1. 1Ratio: comparing parts, readunderstood
  2. 2Ratios and fractions, readunderstood
  3. 3Sharing in a ratio, readunderstood
  4. 4Finding one first, readunderstood
  5. 5Direct proportion, readunderstood
  6. 6Rates and speed, readunderstood
  7. 7Maps and scale drawings, readunderstood
  8. 8Inverse proportion, readunderstood

M3.4.6 Chapter 6

Rates and speed

Priya walks 6 km in 2 hours. Tariq walks 10 km in 5 hours. Who walks faster?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Work out a speed in km per hour from distance and time
  • Find which of two bags gives more rice for each coin

A rate compares two different kinds of amount, such as kilometres and hours. Speed is a rate.

Speed tells you how far you go in each unit of time. A speed in km per hour is the distance in one hour. A speed of 3 km per hour means 3 km in each hour. The word per means for each.

Draw it first

Find the distance for one hour

  1. Step 1: two number lines, time (hours) above and distance (km) below. 0 hours match 0 km, and 2 hours match 6 km. Step 2: an arrow marked ÷ 2 goes to 1 hour and 3 km. Step 3: the label says speed: 3 km per hour.time (hours)distance (km)0026

    1 Priya walks 6 km in 2 hours. Mark both on the number lines.

  2. Step 1: two number lines, time (hours) above and distance (km) below. 0 hours match 0 km, and 2 hours match 6 km. Step 2: an arrow marked ÷ 2 goes to 1 hour and 3 km. Step 3: the label says speed: 3 km per hour.time (hours)distance (km)002613÷ 2

    2 Divide by 2 to find 1 hour: 6 ÷ 2 = 3 km.

  3. Step 1: two number lines, time (hours) above and distance (km) below. 0 hours match 0 km, and 2 hours match 6 km. Step 2: an arrow marked ÷ 2 goes to 1 hour and 3 km. Step 3: the label says speed: 3 km per hour.time (hours)distance (km)002613÷ 2speed: 3 km per hour

    3 Priya walks 3 km in each hour. Her speed is 3 km per hour.

Worked example

Kofi cycles 45 km in 3 hours. What is his speed?

  1. Speed is distance ÷ time.
  2. 45 ÷ 3 = 15.
  3. His speed is 15 km per hour.

Answer: 15 km per hour

Two number lines, time (hours) above and distance (km) below. 0 hours match 0 km, 1 hour matches 15 km, and 3 hours match 45 km.time (hours)distance (km)00115345
Kofi goes 15 km in each hour, so 3 hours is 45 km.

If you know the speed, you can find the distance. Distance = speed × time. You can also find the time. Time = distance ÷ speed.

Rates also help you choose the better buy. Find how much you get for 1 coin. The bag with more rice for 1 coin is the better buy.

Worked example

Bag A holds 5 kg of rice for 2 coins. Bag B holds 9 kg of rice for 4 coins. Which bag gives more rice for each coin?

  1. Bag A, rice for 1 coin: 5 ÷ 2 = 2.5 kg.
  2. Bag B, rice for 1 coin: 9 ÷ 4 = 2.25 kg.
  3. Compare: 2.5 is more than 2.25, so bag A gives more rice for each coin.

Answer: Bag A is the better buy. It gives 2.5 kg for each coin. Bag B gives 2.25 kg.

A table with three columns: bag, rice (kg) and coins. Bag A has 5 kg of rice for 2 coins. Bag B has 9 kg of rice for 4 coins.bagrice (kg)coinsA52B94
Bag B is bigger. But look at the rice for each coin.

See it three ways

With things

Priya walks 6 km in 2 hours. Put 6 stones in 2 equal piles, one pile for each hour.

As a picture

A bar cut into 2 equal parts. The left part is labelled hour 1: 3 km and the right part is labelled hour 2: 3 km. The caption says 6 km in 2 hours.hour 1: 3 kmhour 2: 3 km6 km in 2 hours

In symbols

6 km ÷ 2 hours = 3 km per hour.

Try it

Find your own walking speed

You need: a piece of string 10 m long, or a tape measure, a friend who can count seconds.

  1. Lay the string on the ground to mark 10 m.
  2. Walk along it at a steady pace. Your friend counts the seconds out loud.
  3. Divide 10 by the number of seconds. This is your speed in metres per second.
  4. Walk it again, faster. Compare the two speeds.

What you will see: For example, 10 m in 5 seconds is 10 ÷ 5 = 2 metres each second. A faster walk takes fewer seconds, so its speed number is bigger.

If you cannot try it, this is what happens.
Two number lines, time (seconds) above and distance (m) below. 0 seconds match 0 m, 5 seconds match 10 m, and 10 seconds match 20 m. The speed is 2 m each second.time (seconds)distance (m)005101020

A wrong idea many people have

The bigger bag always gives more rice for each coin.

Why it feels right: A bigger bag holds more rice, so it looks like the better buy.

The test that breaks it: Bag C holds 12 kg for 6 coins. That is 12 ÷ 6 = 2 kg for each coin. Bag D holds 15 kg for 10 coins. That is 15 ÷ 10 = 1.5 kg for each coin. Bag D is bigger, but bag C gives more for each coin.

What is true: Divide the amount by the number of coins for each bag. The bag with the bigger answer is the better buy.

Key words

rate
a comparison of two different kinds of amount, such as km and hours.
speed
the distance you go in each hour, or each second, such as 15 km per hour.
per
for each, as in 3 km per hour.

Back to your guess

The answer is B: Priya, because she walks 3 km in each hour and Tariq walks 2 km.

To compare speeds, find the distance for 1 hour. Priya walks 6 ÷ 2 = 3 km each hour. Tariq walks 10 ÷ 5 = 2 km each hour. Priya is faster, even though Tariq goes further.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A train travels 360 km in 4 hours. What is its speed in km per hour?

    Two number lines, time (hours) above and distance (km) below. 0 hours match 0 km, 1 hour matches a question mark, and 4 hours match 360 km.time (hours)distance (km)001?4360
    Tap your answer.

    1. Not this one. You multiplied 360 by 4. Speed is distance divided by time, not multiplied. Go to M3.4.6. Try another.

    2. Yes, that is right. 360 ÷ 4 = 90. The train goes 90 km in each hour.

    3. Not this one. You took 4 away from 360. Speed is the distance for 1 hour, so divide 360 by the 4 hours. Go to M3.4.6. Try another.

    See the answer

    90 km per hour

    360 ÷ 4 = 90. The train goes 90 km in each hour.

    • If you chose A: You multiplied 360 by 4. Speed is distance divided by time, not multiplied. Go to M3.4.6.
    • If you chose C: You took 4 away from 360. Speed is the distance for 1 hour, so divide 360 by the 4 hours. Go to M3.4.6.

  2. A bus drives at 45 km per hour for 3 hours. How far does it go?

    Two number lines, time (hours) above and distance (km) below. 0 hours match 0 km, 1 hour matches 45 km, and 3 hours match a question mark.time (hours)distance (km)001453?
    Tap your answer.

    1. Not this one. You added the speed and the time. The bus goes 45 km in each of 3 hours, so multiply. Go to M3.4.6. Try another.

    2. Not this one. You divided 45 by 3. To find a distance, multiply the speed by the time. Go to M3.4.6. Try another.

    3. Yes, that is right. The bus goes 45 km in 1 hour. In 3 hours it goes 3 × 45 = 135 km.

    See the answer

    135 km

    The bus goes 45 km in 1 hour. In 3 hours it goes 3 × 45 = 135 km.

    • If you chose A: You added the speed and the time. The bus goes 45 km in each of 3 hours, so multiply. Go to M3.4.6.
    • If you chose B: You divided 45 by 3. To find a distance, multiply the speed by the time. Go to M3.4.6.

  3. A cyclist rides at 15 km per hour. How many hours does it take to ride 60 km?

    Two number lines, time (hours) above and distance (km) below. 0 hours match 0 km, 1 hour matches 15 km, and a question mark of hours matches 60 km.time (hours)distance (km)00115?60
    Tap your answer.

    1. Not this one. You multiplied 15 by 60. Time is distance divided by speed. Ask how many lots of 15 km are in 60 km. Go to M3.4.6. Try another.

    2. Not this one. You took 15 away from 60. Ask how many lots of 15 km fit into 60 km, so divide. Go to M3.4.6. Try another.

    3. Yes, that is right. Each hour is 15 km. 60 ÷ 15 = 4, so it takes 4 hours.

    See the answer

    4 hours

    Each hour is 15 km. 60 ÷ 15 = 4, so it takes 4 hours.

    • If you chose A: You multiplied 15 by 60. Time is distance divided by speed. Ask how many lots of 15 km are in 60 km. Go to M3.4.6.
    • If you chose B: You took 15 away from 60. Ask how many lots of 15 km fit into 60 km, so divide. Go to M3.4.6.

  4. Bag P holds 9 kg of rice for 3 coins. Bag Q holds 14 kg of rice for 7 coins. Which bag gives more rice for each coin?

    A table with three columns: bag, rice (kg) and coins. Bag P has 9 kg of rice for 3 coins. Bag Q has 14 kg of rice for 7 coins.bagrice (kg)coinsP93Q147
    Tap your answer.

    1. Not this one. Bag Q holds more rice, but it also takes more coins. Divide the rice by the coins for each bag. Go to M3.4.6. Try another.

    2. Yes, that is right. Bag P gives 9 ÷ 3 = 3 kg for each coin. Bag Q gives 14 ÷ 7 = 2 kg for each coin. 3 is more than 2.

    3. Not this one. Check each bag: 9 ÷ 3 and 14 ÷ 7. Are the answers the same? Go to M3.4.6. Try another.

    See the answer

    Bag P

    Bag P gives 9 ÷ 3 = 3 kg for each coin. Bag Q gives 14 ÷ 7 = 2 kg for each coin. 3 is more than 2.

    • If you chose A: Bag Q holds more rice, but it also takes more coins. Divide the rice by the coins for each bag. Go to M3.4.6.
    • If you chose C: Check each bag: 9 ÷ 3 and 14 ÷ 7. Are the answers the same? Go to M3.4.6.

  5. Priya walks 12 km in 3 hours. Tariq walks 10 km in 2 hours. Who walks faster?

    A table with three columns: name, km and hours. Priya walks 12 km in 3 hours. Tariq walks 10 km in 2 hours.namekmhoursPriya123Tariq102
    Tap your answer.

    1. Not this one. Priya goes further, but she takes 3 hours. Compare the distance for each hour, not the whole distance. Go to M3.4.6. Try another.

    2. Yes, that is right. Priya: 12 ÷ 3 = 4 km per hour. Tariq: 10 ÷ 2 = 5 km per hour. 5 is more than 4, so Tariq is faster.

    3. Not this one. Check each speed: 12 ÷ 3 and 10 ÷ 2. Are they the same? Go to M3.4.6. Try another.

    See the answer

    Tariq, at 5 km per hour. Priya goes 4 km per hour.

    Priya: 12 ÷ 3 = 4 km per hour. Tariq: 10 ÷ 2 = 5 km per hour. 5 is more than 4, so Tariq is faster.

    • If you chose A: Priya goes further, but she takes 3 hours. Compare the distance for each hour, not the whole distance. Go to M3.4.6.
    • If you chose C: Check each speed: 12 ÷ 3 and 10 ÷ 2. Are they the same? Go to M3.4.6.

  6. A cyclist rides 60 km in 4 hours. What is the speed in km per hour?

    See the answer

    15 km per hour

    60 ÷ 4 = 15. The cyclist goes 15 km in each hour.

  7. A walker goes at 4 km per hour. How many hours does it take to walk 20 km?

    See the answer

    5 hours

    Each hour is 4 km. 20 ÷ 4 = 5, so it takes 5 hours.

  8. From an earlier chapter

    Two amounts are in direct proportion. When the first is 5, the second is 15. What is the second when the first is 8?

    See the answer

    24

    15 ÷ 5 = 3, so the second is always 3 times the first. 8 × 3 = 24.

Keep it

Copy this into your exercise book.

Speed is distance ÷ time, in km per hour. Distance = speed × time. For the better buy, find the amount for 1 coin. The bigger bag is not always better.

Draw: Draw two number lines. Mark 2 hours and 6 km. Draw an arrow to 1 hour and 3 km. Write 3 km per hour.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (710 KB). It opens on any phone.