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Ratio, rate and proportion

What is a ratio?

A ratio is a comparison of two or more numbers that are usually of the same type or measurement. If the numbers have different units, it is important to convert the units to be the same before doing any calculations.

We write the numbers in a ratio with a colon (:) between them.

For example, if there are 8 learners who travel by bus and 12 learners who travel by taxi, then we say we have a ratio of 8 learners travelling by bus to 12 learners travelling by taxi.

We can write this as 8 : 12. We can also simplify this ratio to 2 : 3, by dividing both parts by 4.

It is important in which order you state the ratio. A ratio of 1 : 7 is not the same as a ratio of 7 : 1.

Note:

Ratios don't have measurement units, because the units cancel out. So we write a ratio of 3 litres to 4 litres as 3 : 4, without writing 'litres'. The units only cancel out if they are the same, though! For example, a ratio of 300 ml to 1 litre must always be written as 300 : 1000 before we can simplify it to 3 : 10.

Writing ratios in the simplest form and equal ratios

You can write a ratio in its simplest form in the same way as you would write a fraction in its simplest form. Check if there is a number that divides into both numbers, starting with the smallest number in the ratio, and then checking with 2; 3; 5; etc. If there is none, then the ratio is already in its simplest form.

To check if ratios are equivalent write both of them in their simplest form, which will be exactly the same if they are equal. For example 5 : 10 and 30 : 60 are equivalent ratios because they both simplify to 1 : 2.

Example 1: Writing ratios in the simplest form

Question

Write these ratios in their simplest forms:

  1. 5 : 30
  2. 14 : 18
  3. 18 : 30
  4. 7 : 280
Answer
  1. 5 and 30 are both divisible by 5. So the ratio simplifies to 1 : 6.
  2. 14 and 18 are both divisible by 2. So the ratio simplifies to 7 : 9.
  3. 18 and 30 are both divisible by 3. So the ratio simplifies to 6 : 10. However, these two numbers can both be divided by 2. So it simplifies further to 3 : 5.
  4. 7 and 280 are both divisible by 7. So the ratio simplifies to 1 : 40.

Writing ratios in unit form

Writing a ratio in the simplest form will sometimes result in one of the numbers being equal to 1. This is called a unit ratio. For example, the ratio of 5 lillies to 15 daisies in a bunch of flowers is simplified to 1 : 3.

In some situations a unit ratio is not in the simplest form, for example, 5 : 9 can be written as 1 : 1,8, which is a unit ratio, but not in the simplest form. To calculate the unit form, we simply divide both numbers by the smaller number, so \(\text{5} \div \text{5} : \text{9} \div \text{5} = \text{1} : \text{1,8}\).

Let's look at some situations in which the unit ratio is useful.

Example 2: Writing ratios in the unit form

Question
  1. There are 23 nurses in a hospital and 7567 patients. How many patients does each nurse have to care for?
    Image
  2. In a Grade 10 class, learners are voting for a class badge. 4 learners vote for badge A and 17 vote for badge B. How many learners vote for badge B for each learner voting for badge A?
Answer
  1. 23 : 7567.
    Divide both numbers by 23:

    \(\text{23} \div \text{23} : \text{7567} \div \text{23} = \text{1} : \text{329}\)

    So each nurse has 329 patients to care for on average.

  2. 4 : 17.
    Divide both numbers by 4.

    \(\text{4} \div \text{4} : \text{17} \div \text{4} = \text{1} : \text{4,25}\)

    So there are 4,25 votes for Badge B for every one vote for Badge A.

Exercise 1: Working with ratios

Which of these pairs of ratios are equal?

  1. 3 : 4 and 75 : 10

  2. 2 : 3 and 10 : 20

  3. 5 : 1 and 100 : 20

  4. 10 : 1 and 40 : 5

  1. Not equal

  2. Not equal

  3. Equal

  4. Not equal

The ratio of female learners to male learners in a class is 3 : 2. If there are 30 female learners in the class, work out:

  1. the number of male learners

  2. the total number of learners in the class

  1. 3 : 2 is equal to 30 : 20 so there are 20 male learners.

  2. 20 + 30 learners = 50 learners

A fruit and nut company has the following standards requirement: In a packet of dried fruit and nuts, there must be two hundred grams of fruit for every 50 g of nuts.

  1. Write this as a simple ratio.

  2. What will the amount of fruit be if there are 500 g of nuts?

  3. What will the amount of fruit be if there are 25 g of nuts?

  1. 200 : 50

  2. 200 : 50 is equal to 2000 : 500, so there will be 2000 g of fruit and 500 g of nuts

  3. 200 : 50 is equal to 100 : 25 so there will be 100 g of fruit and 25 g of nuts.

Tshepo wants to make orange juice out of concentrated juice. The bottle says that it must be diluted 1 : 7 with water. If he wants to make 2 litres (2000 ml) of juice in total, how many millilitres of water must he mix with how much of the concentrated juice?

1 plus 7 parts equals 8 parts in total. \(\text{2000} \div\) by 8 is 250 ml. 1 : 7 is equal to 250 : 1750. So he must mix 250 ml of concentrate with 1750 ml of water.

What is a rate?

A rate, like a ratio, is also a comparison between two numbers or measurements, but the two numbers in a rate have different units.

Some examples of rate include cost rates, (for example potatoes cost R 16,95 per kg or 16,95 R/kg) and speed (for example, a car travels at 60 km/h).

Image

When we calculate rate, we divide by the second value, so we are finding the amount per one unit.

Unit rates

For example, if we want a rate for R 20 for 2 kg of flour, we write:

R 20 : 2 kg = R 10 : 1 kg

= R 10/kg.

This rate is a unit rate.

Example 3: Calculating rates

Question
  1. Elias, a star athlete, runs 100 m in 15 seconds.

    1. What is his speed in metres per second?
    2. If he was able to keep running at this speed, how long would he take to cover 1 km?

  2. Cheese costs R 56 per kg. Thandi buys 200 g of cheese. How much does she pay?
    Image
Answer
    1. 100 m \(\div\) 15 sec = 6,67 m/sec
    2. 100 seconds or 1 min 40 sec
  1. There are 1000 g in 1 kg. We can either reason that there are 5 \(\times\) 200 g in the 1000 g, and divide the cost by 5, or we can work out the cost per 100 g and then work out how much 200 g costs. The cost per 100 g would be R 5,60, so 200 g costs R 11,20.

Exercise 2: Working with rates

A packet of 6 handmade chocolates costs R 15,95. How much does each chocolate cost?

Image

\(\text{R 15,95} \div \text{6} = \text{R 2,668} \ldots\) rounded off to R 2,67 per chocolate.

A truck driver travels 1500 km in 18 hours. What is his average speed?

\(\text{1500}\text{ km} \div\) 18 h = 83,33\(\ldots\) km/h. Round off to 83,33 km/h.

Nicola is able to input 96 words in 2 minutes on her laptop. Karen times her own typing speed as 314 words in 7 minutes. Work out their speeds to see who is faster.

Nicola types 96 words \(\div\) 2 min = 48 words/min. Karen types 314 words \(\div\) 7 min = 44,857\(\ldots\) words/min. Round off to a whole word: 45 words/min. Nicola is faster.

Finding missing numbers in ratios and rates

Solving problems often involves using ratios and rates to find unknown values. We use a similar process to find missing numbers in ratios and rates.

Example 4: Finding missing numbers in a ratio

Question

Thenji makes a fruit salad for breakfast at a restaurant. She uses pieces of fruit in the following ratio:

banana : apple : paw-paw

1 : 2 : 3

  1. If she uses 20 pieces of apple, how many pieces of banana and paw-paw should she use?
  2. If she uses 12 pieces of banana, how many pieces of apple and paw-paw should she use?

Image

Answer

  1. Let's start with the banana : apple ratio.
    1 : 2 = banana pieces : apple pieces
    1 : 2 = 10 : 20
    So we have 10 banana pieces and 20 apple pieces.
    Next we look at the apple : paw-paw ratio.
    2 : 3 = apple pieces : paw-paw pieces.
    2 : 3 = 20 : 30
    So we have 20 apple pieces and 30 paw-paw pieces.
    So the complete ratio will be 10 : 20 : 30.
  2. Let's start with the banana : apple ratio.
    1 : 2 = banana pieces : apple pieces.
    1 : 2 = 12 : 24
    So we have 12 banana pieces and 24 apple pieces.
    Next, we look at the apple : paw-paw ratio.
    2 : 3 = apple : paw-paw.
    2 : 3 = 24 apple pieces : ? paw-paw pieces.
    Write in fraction form: \(\frac{\text{2}}{\text{3}} = \frac{\text{24}}{\text{paw-paw}}\)
    One way of solving this is to cross-multiply:
    So paw-paw pieces \(\times \text{2} = \text{3} \times \text{24}\)
    therefore paw-paw pieces = \(\text{72} \div \text{2} = \text{36}\) pieces
    So the complete ratio will be 12 : 24 : 36.

Direct proportion and inverse proportion

Equal ratios have a directly proportional relationship. There is another kind of proportion that we need to investigate.

  • Direct proportion: as one quantity increases, the other increases OR as one quantity decreases, the other decreases.
  • Inverse proportion: as one quantity decreases, the other increases OR as one quantity increases, the other decreases.

Example 5: Working with inverse proportion

Question

The learners at a school want to hire a hall to hold a party. They can hire the use of a hall for one evening for R 3000. The learners who are going to the party need to split the cost between them.

  1. Draw up a table to show the cost per learner if 30; 50; 100; 200 and 300 learners attend the party.
  2. The learners decide that they can't hold the party if they need to pay more than R 25 each. What number of learners must go to the party for it to be affordable?
    Image

Answer

  1. Table 1

    Number of learners

    Cost per learner going to the party

    30

    100

    50

    60

    100

    30

    200

    15

    300

    10

  2. \(\text{R 3000} \div \text{R 25} = \text{120}\) learners. So there must be at least 120 learners going to the party.

Exercise 3: Finding unknown values in ratios and rates

For the following problems, calculate the unknown values. The letter \(x\) indicates an unknown value.

  1. 5 hats are to 4 coats as \(x\) hats are to 24 coats.

  2. \(x\) cushions are to 2 couches as 24 cushions are to 16 couches.

  3. 1 spacecraft is to 7 astronauts as 5 spacecraft are to \(x\) astronauts.

  4. 18 calculators are to 90 calculators as \(x\) students are to 150 students.

  5. \(x\) TV's are to R 40 000 as 1 TV is to R 1000.

  1. \(x=\text{30}\)

  2. \(x = \text{3}\)

  3. \(x=\text{35}\)

  4. \(x =\text{30}\)

  5. \(x = \text{40}\)

Indicate whether the following proportions are true or false:

  1. \(\frac{\text{3}}{\text{16}}=\frac{\text{12}}{\text{64}}\)

  2. \(\frac{\text{2}}{\text{15}}=\frac{\text{10}}{\text{75}}\)

  3. \(\frac{\text{1}}{\text{9}}=\frac{\text{3}}{\text{30}}\)

  4. \(\frac{\text{6} \text{ knives}}{\text{7} \text{ forks}}=\frac{\text{12}\text{ knives}}{\text{15} \text{ forks}}\)

  5. \(\frac{\text{33}\text{ kilometres}}{\text{1}\text{ litre}}=\frac{\text{99} \text{ kilometres}}{\text{3} \text{ litre}}\)

  6. \(\frac{\text{320}\text{ metres}}{\text{5} \text{ seconds}}=\frac{\text{65}\text{ metres}}{\text{1}\text{ second}}\)

  7. \(\frac{\text{35} \text{ students}}{\text{70}\text{ students}}=\frac{\text{1}\text{ class}}{\text{2} \text{classes}}\)

  1. True

  2. True

  3. False

  4. False

  5. True

  6. False

  7. True

Write the simplified form of the rate “sixteen sentences to two paragraphs.”

\(\frac{\text{8} \text{ sentences}}{ \text{1} \text{ paragraph}}\)

A rectangle has a fixed area of 81 square units.

  1. Complete the table to show the inverse proportion relationship between the length and the breadth of the rectangle:

    Table 2

    length (cm)

    1

    3

    9

    27

    81

    breadth (cm)

    3

  2. If this rectangle is to be used as a serviette, which of the measurements are reasonable?

  1. Table 3

    length (cm)

    1

    3

    9

    27

    81

    breadth (cm)

    81

    27

    9

    3

    1

  2. \(\text{9}\times\) 9 units