Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
UNIT 8: NUMERICAL PROPORTIONALITY.
Ratio.
A ratio is a comparison of two numbers.
Suppose we want to write the ratio of 8 and 12.
We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.
The ratio of two numbers a and b is the quotient
a
.
b
Example:
Susana has a bag with 3 pens, 4 pencils, 7 felt tip pens, and 1 pencil sharpener. What is the ratio of
pencils to pens?
Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to
the second, the answer would be 4/3.
Two other ways of writing the ratio are 4 to 3, and 4:3.
Proportion
An equation in which two ratios are equal is called a proportion.
Two ratios
a
and
b
c
d
a c
=
.
b d
are a proportion when
Remember the two ratios are equals if their cross products are equal a⋅d =b⋅c .
You can also use the cross product to find a missing number:
Example:
5 x
=
.
2 8
1. Solve:
5⋅8=2x ⇒ 40=2x ⇒ 2x=40 ⇒ x=
2. Find x to get a proportion
40
⇒ x=20
2
10 x
= .
12 6
}
10 x
60
= ⇒12x=10 · 6⇒ 12x=60 ⇒ x= =5 ⇒ x =5
12 6
12
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
Find x to get a proportion:
a)
15 3
=
25 x
b)
3 x
=
4 20
c)
2 14
=
x 21
d)
x 4
=
45 9
Direct proportion
Two quantities are in direct proportion when they increase or decrease in the same ratio.
Example:
We want to paint our farm so we need to use the following paint mixture: four green pots and one
pot white. The ratio is 4 pots green to 1 pot white, or 4:1.
If we also want to paint our neighbour's farm, which it is two times ours, we have to double the
amount of paint and increase it in the same ratio.
We double the amount of green paint: 8 pots.
We double the amount of white paint: 2 pots.
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
The amount of green and white paint we need increase in direct proportion to each other.
Understanding direct proportions can help you to work out the value or amount of quantities either
bigger or smaller than the one about which you have information.
How to solve proportions using the unitary method
Example:
If you know the cost of 4 light bulbs is £6.00, can you work out the cost of 3 light bulbs?
To solve this problem we need to know the cost of 1 light bulb.
If four light bulbs cost £6.00, then you divide £6.00 by 4 to find the price of 1 light bulb.
6÷4=2,5
Now you know that they cost £1.50 each, to work out the cost of 3 light bulbs you multiply £1.50
by 3. 1,5· 3=4,5
So, 3 light bulbs cost £4.50 .
How to solve proportions using the cross-multiply and divide method.
Example:
The cost of 3 pens is 4.5 €. Work out the cost of 7 pens.
At the fishmarket a fishmonger sell fresh anchovies at 4 kg. for 3 euros. If we have 9 euros, what
quantity of anchovies can we buy?
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
Inverse proportion
Inverse proportion is when one value increase as the other value decreases.
Example:
Think of how long it would take 5 worker to build a building. How long would it take if they had 5
workers helping them? How long would it take if they had ten workers helping you?
The relation between the time it takes to build a building and the number of people working is an
inverse proportion. The more people you have, the less time it takes.
If you double the number of workers, the time is halved.
If you treble the number of workers, the time becomes to a third.
The cross-multiplication method
We are going to apply the same method that we used to solve direct proportions to inverse
proportions, but with one additional step to the process: flip over the second column numbers to
make it work.
Example:
A farmer has enough cattle feed to feed 300 hens for 20 days. If he buys 100 more hens, how long
would the same amount of feed feed the total hens?
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
4 taps can fill a larger container in 6 hours. If we use 12 taps, how long will it take to fill the
container?
Activities.
1. If two pencils cost $1.50, how many pencils can you buy with $9.00?
2. Jane ran 100 meters in 15 seconds. How long did she take to run 1 meter?
3. A car travels 125 miles in 3 hours. How far would it travel in 5 hours?
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
4. It takes 4 men 6 hours to repair a road. How long will it take 7 men to do the job if they work at
the same rate?
5. If you know the cost of 3 packets of battery is ₤6.00, can you work out the cost of 5 packets?
6. 24 people can construct a house in 15 days. But the owner would like to finish the work in 12
days. How many more workers should he employ?
7. Some people working at the rate of 6 hours a day can complete the work in 19 1/2 days. As they
have received another contract, they want to finish this work early. Now they start working 6 1/2
hours a day. In how many days will they finish this work?
8. The groceries in a home of 4 members are enough for 30 days. If a guest comes and stays with
them, how many days will the groceries last?
9. A map scale is 1:25 000. On the map the distance between two shopping centres is 4 cm. What is
the actual distance between the shopping centres? Give your answer in km.
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
Percentages.
A percent is really a fraction with a denominator of 100. The percent symbol, % is another way of
writing a denominator of 100.
In other words: 60% =
60
or 3% =
100
3
.
100
Calculate the % of a given number.
Example:
Find the 85% of 20:
85% of 20 =
85
85⋅20
⋅20=
=17
100
100
Calculate:
The 95% of 900
The 20% of 150
The 30% of 60
The 5% of 400
Determine the percentage (%)
Example
If a drinks bill of £18 for a party costing £30 in total, which percentage of the total did the drinks
cost?
18
1800
100=
=60 %
30
30
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
Complete:
The …......... % of 400 is 380
The ….......... % of 500 is 100
The ….......... % of 800 is 80
The …......... % of 60 is 3
Percentage increases. Tax problems.
Example:
A new iPhone costs $300. If there is a tax rate of 8%, how much do you have to spend in order to
walk out the door with a new phone (including tax)?
original + tax = total
300+0.8(300) = 300+24 = 324
The price of a certain model of TV goes up by 10%. It used to cost £150. What will it cost in
future?
150
10
· 150=15015=165
100
If 700 goes up by 50 % the result will be: ….........
If 120 goes up by 10 % the result will be: ….........
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
Percentage decreases.
Example:
In a sale, there is a rack of trousers marked "All prices in this rack are reduced by 20%!".
The one I choose now had a price of £50. How much have I to pay now?
50−
20
· 50=50−10=40
100
If 800 is reduced by 40% the result will be: ….........
If 150 is reduced by 25% the result will be: ….........
Calculate the initial capital after a percentage increase or decrease
Example:
In a sale, there is a rack of blouses marked "All prices in this rack are reduced by 20%!".
The one I choose now has a discount of £45. How much did it cost before the sale?
20
20x
4500
x=45⇒
=45⇒ 20x=4500⇒ x=
=225
100
100
20
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
Activities.
1. In a survey of Juan's class of 35 students, 80% said they liked the sandwich of the bar of their
school. How many students liked the sandwich of the bar?
2. Diana has played on the football team 40 times this year. She has gotten a goal in 35 of those
games. In what percent of the games that she's played this year has Diana gotten a goal?
3. Mr. Lopez has 25 coins in his pocket, of which 11 are coins of one euro. What percent of the
coins are of one euro?
4. Of 200 students of first E.S.O., 85% scored 6,3 or higher in a mathematics test. How many
students scored 6,3 or more?
5. Of 50 teachers, only 12% do not exercise. What number of teachers do exercise?
6. In a field, there are 24 horses and 36 cows. What percentage of those animals are horses?
7. The instruments owned by the marching band are 34% trombones, 24% drums and 42%
saxophones. If the band owns 50 instruments in total, how many of each musical instrument are
there?
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Unit 8: Numerical Proportionality. Mathematics 1st E.S.O.
Teacher: Miguel Angel Hernández
8. The price of a book is on sale, 12% cheaper. If the book's price was €10, how much does the
book cost now?
Mathematics Glossary
Write here new words you learned related to mathematics:
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