This document provides instruction on ratios, proportions, and solving related math problems. It begins with defining ratios and providing examples of simplifying and writing ratios in simplest form. It then discusses using ratios to solve word problems by calculating total amounts. The document also covers direct and inverse proportions, providing examples of each. It explains how to set up and solve proportion word problems involving variables like time, workers, or other quantities that have a direct or inverse relationship. Several practice questions are provided to allow working through example proportion problems.

1. FUNCTIONAL
SKILLS – Math

2. Today!
◦ Starter: Fraction Fun!
◦ Tutorial: Percentages – Sub
topics 1 to 5
◦ Break!
◦ Tutorial continued
◦ Activity
◦ Extended work, if time:
practice-sheets

3. Ratio

6. Ratio
A ratio is something we use to compare how much there is of one thing to how much there is of another.
A ratio is written like 2:5 and is spoken as “2 to 5”. This means that for every 2 lots of one thing, there must
be 5 lots of the other.
Simplifying Ratios
To simplify a ratio, divide each part of the ratio by the same number each time, until
they can no longer be divided.
Example: Write the ration 18:60:24 in its simplest form.
We can divide each number by 3 and then by 22 (or just by 6)
This cannot be simplified anymore to make whole numbers, so the ratio is in its simplest form, which 3:10:4

7. Solving ratio questions

8. PRACTICE QUESTION:
Olive makes her tea by adding 1 part milk to 7 parts hot water (1:7). If Olive uses 30 ml of milk, how much hot
water does she use?

10. You can use ratios to calculate total amounts, using the following steps:
Step 1: Calculate the value of one part (you may be given this in the question).
Step 2: Calculate the total number of parts.
Step 3: Calculate the total amount, by multiplying the value of one part by the total number of parts.
Example: A basic dough is made by mixing 3 parts of Greek yoghurt and 4 parts self-raising flour.
160 g of self-raising flour is used. How much dough is made in total?
4 parts self-raising flour is 160 g, so
1 part=160÷4=40 g
The total number of parts is
3+4=7
So, the total amount of dough made is 7×40g = 280 g
Practice Question:
Adam, Ben and Charlie are three brothers. The brother’s ages added together is 63. The ratio of their ages
is 3:4:2. How old is each brother?

11. For each of these, find the ratio in its simplest form.
1. On a bus there are 24 seats upstairs and 27 seats downstairs. What is the ratio of the number of seats
upstairs to the number of seats downstairs?
2. One morning a postman delivered 42 first-class letters and 48 second-class letters. What is the ratio of the
number of first-class letters to the number of second-class letters?
For each of these, find the quantities.
Water in a swimming pool is treated with two chemicals mixed in the ratio of 5 : 4. The total volume of the
chemicals is 48 litres. How much of each chemical is used?
3. Concrete consists of six parts gravel to one part cement. A builder makes up 140 kg of concrete mixture.
What is the weight of gravel used in this mixture?
4. A drink is made from juice and water in the ratio of 1:5. How many litres of drink can be made from 2
litres of juice?
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12. Proportion
Two quantities are proportional if, as one changes, the other changes in a certain way.
We will explain direct proportion and inverse proportion.
3

14. Direct Proportion
Two quantities are directly proportional if as one increases, the other
one increases at the same rate, e.g. as one is doubled, the other is doubled.
Example: Toni uses 150 g of chocolate to make 6 cookies. How much chocolate
would Toni need to make 20 cookies?
Step 1: Divide the amount of chocolate by 6 to find the amount needed for 1 cookie.
150÷6=25 g of chocolate
Step 2: Multiply the amount of chocolate needed for 1 cookie by the 20 cookies
needed.
20×25=500 g of chocolate

17. Inverse Proportion
Two quantities are inversely proportional if as one increases, the other
one decreases at the same rate, e.g. as one is doubled, the other is halved.
Example: It takes 8 workers 25 months to build 10 houses. Assuming they all work
at the same rate, how long would it take 20 workers to build the same number of
houses?
Step 1: Multiply the number of workers by the number of months, to find the time it
would take 1 worker to build 10 houses:
8×25=200 months
Step 2: Divide the time it takes 1 worker to build 10 houses by 20 workers, to get
the answer:
200÷20=10 months

18. Example: speed and travel time
Speed and travel time are Inversely Proportional because the faster we go the shorter
the time.
As speed goes up, travel time goes down
And as speed goes down, travel time goes up
‘y’ is inversely proportional to ‘x’
Is can be written as: y = 1/x

19. Example: 4 people can paint a fence in 3 hours. How long will it take 6 people to paint it?
(Assume everyone works at the same rate)
It is an Inverse Proportion:
•As the number of people goes up, the painting time goes down.
We can use:
t = k/n
Where:
•t = number of hours
•k = constant of proportionality
•n = number of people
"4 people can paint a fence in 3 hours" means that t = 3 when n = 4
3 = k/4
3 × 4 = k × 4 / 4
k = 12
So now we know:
t = 12/n
And when n = 6:
t = 12/6 = 2 hours
So 6 people will take 2 hours to paint the fence.

20. Question 1: It takes 60 minutes for 3 gardeners to cut the grass of a field. Assuming they
all work at the same rate, how long would it 5 gardeners to cut the same grass?
Question 2: It takes 4 builders 6 weeks to build a structure. How long would it have
taken 6 builders to build the same structure?
Question 3: It takes 3 gardeners 45 minutes to cut the grass of a sports field. Assuming
they all work at the same rate, how long would it take:
(a) 5 gardeners to cut the same grass?
(b) 2 gardeners to cut the same grass?
Test your Skill

24. GAME TIME
https://www.mathplayground.com/ASB_RatioBlaster.html