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Per means for each. A phrase that contains the word
‘per’ is called a RATE.
Example : Sean walks at 5 km per hour (km/hr)
This means that if Sean walked at this speed forone hour he would travel 5km.
RatesRates
We are often interested in rates
•Miles per gallon•Goals per game•£’s spent per day•Calories per day•Words per minute
Rates
Why are we interested in rates?
Example : Sean walks at 5 km per hour (km/hr)
How far would he walk in :
(a) 4 hours (b) Half an hour
(a)
20km
RatesRates
1 5
4 4 x 5 =
Hours Distance walked
(b)
2.5km1 5
0.5 0.5 x 5 =
Hours Distance walked
Finding the Rate
Example : For 5 hours’ work Jennifer is paid £30.Calculate her rate of pay per hour.
Answer :
5 hours £30
1 hour £30 ÷ 5 =£6
This means that Jennifer is paid a ‘rate’ of£6 per hour
Finding the Rate
Example : Nicola gets paid £48 000 pounds a year. What is her monthly rate of pay.
Answer :
12 months £48 0001 month £48 000 ÷ 12 =£4 000
This means Nicola gets paid £4 000 per month
Direct Direct ProportionProportion
“ .. When you double the number of cakes you double the cost.”
Cakes Cost
Two quantities, (for example, number of cakes and total cost) are said to be in DIRECT PROPORTION, if :
Example : The cost of 6 cakes is £4.20. find the cost of 5 cakes.
6 4.20 1 4.20 ÷ 6 = 0.70 5 0.70 x 5 = £3.50
Direct Direct ProportionProportion
£ $
Example : On holiday I exchanged £30 for $45.How many $ will I get for £50.
30 45 1 45 ÷ 30 = 1.5 50 1.5 x 50 = $75
What name do we give to
this value
Exchange rate
Direct Direct ProportionProportion
Pencils Cost
Sometimes it is easier to find the cost of 10,100 or 1000 items rather than 1.
Example : 300 pencils cost £6. How much will 200 cost.
300 £6.00 100 £6.00 ÷ 3 = £2.00200 £2.00 x 2 = £4.00
ProportionProportion
Men Hours
Inverse proportion is when one quantity increasesand the other decreases. The two quantities are said
to be INVERSELY PROPORTIONAL or (INDIRECTLY PROPORTIONAL) to each other.
Example : If it takes 3 men 8 hours to build a wall.How long will it take 4 men.
3 8 1 4
Inverse Proportion
Are we expecting more or
less
Less Time !!!
3 x 8 = 2424 ÷ 4= 6 hours
1 8
ProportionProportion
Men Months
Example : It takes 10 men 12 months to build a house.How long should it take 8 men.
10 12
Inverse Proportion
Are we expecting more or
less
More time !!!
12 x 10= 120
120 ÷ 8= 15 months
Ratios can be used to compare different quantitiesRatios can be used to compare different quantities
Example : There are 2 triangles and 3 rectangles.
The ratio of triangles to rectangles is said to be 2 : 3
Note: The ratio of rectangles to triangles is said to be 3 : 2
RatioRatioSimplifying a ratio ?
Simplifying a ratio is like simplifying fractionsSimplifying a ratio is like simplifying fractions
Fraction :÷2
÷2
68
Ratio : ÷2 ÷26 : 8
RatioRatioSimplifying a ratio ?
68 = =
34
6 : 8 = =3 : 4
Divide top and bottom by highest common
factor
Ratio Calculations
Example : The ratio of boys to girls is 4:5. If there are 16 boys, how many girls are there.
boys girls
4 5
16 20x 4 x 4
RatioRatio
Ratio Calculations
Example : The ratio of cars to buses is 3:7. If there are 49 buses, how many cars are there?
cars buses
3 7
21 49x 7 x 7
RatioRatio
Ratio CalculationsRatioRatio
Example : The ratio of goals to games for a footballer is 1 : 4. If he plays52 games, how many goals would he be expected to score?
goalsgames
1 4
13 52x 13 x 13
Ratio & Ratio & ProportionProportion
Example : Bill and Ben share a raffle win of £400 in the ratio 3:5. How much does each get ?
Proportional Division
Step 1 : Since the ratio is 3:5, there are :
3+5 = 8 shares
Step 2 : Each share is worth : 50
8 400
Step 3 : Bill gets 3 x 50 = £150
Ben gets 5 x 50 = £250Check !150 + 250 = 400
Ratio & Ratio & ProportionProportion
Example : Ryan and Ross share 24 cakes in the ratio3:1. How many cakes does each get ?
Proportional Division
Step 1 : Since the ratio is 3:1, there are :
3+1 = 4 shares
Step 2 : Each share is worth : 6
4 24
Step 3 : Ryan gets 3 x 6 = 18
Ross gets 1 x 6 = 6Check !18 + 6
= 24
