Numerical Proportionality

Mathematics- 4th Secondary Option A

NUMERICAL

PROPORTIONALITY

1. RATIOS AND PROPORTIONS 2. DIRECTLY PROPORTIONAL MAGNITUDES 3. INVERSELY PROPORTIONAL MAGNITUDES 4. PERCENTAGES 5. MIXTURES 6. SIMPLE AND COMPOUND INTEREST 7. SOLVED PROBLEMS 1. RATIOS AND PROPORTIONS The ratio between two numbers a and b is the fraction a/b. Ratios can be shown in different ways: 3:4

Using the “:” to separate example values

¾

as a fraction, by dividing one value by the total

0.75

as a decimal

75%

as a percentage

A proportion is an equation that states that two ratios are equal. It is a statement that two ratios are equal. 3 6 is an example of a proportion =

4

8

RULE: In a true proportion, the product of the means equals the product of the extremes. 2. DIRECTLY PROPORTIONAL MAGNITUDES Two quantities are in direct proportion when they increase or decrease in the same ratio. For example, you could increase something by doubling it, or decrease it by halving. If two values x and y are directly proportional to each other then the ratio x : y or x/y is a constant (i.e. always remains the same). When two magnitudes are directly proportional, if you multiply or divide by the same number a pair of corresponding values, you get another pair of corresponding values. When we are trying to find a number in an exercise involving magnitudes in a direct proportion there are two methods:

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Maths Department / I.E.S Al-qázeres

Numerical Proportionality

•

Mathematics- 4th Secondary Option A

I -Unitary method 1. We convert the proportion in 1:n or n:1 (the most convenient) 2. We multiply by the third quantity.

Example : A man walks 5200m in 2,5 hours. How much will he walk in 7 h at the same speed? 1. If he walks 5200 m in 2.5 h, in 1 h he will walk 5200 / 2 .5 = 2080 m . 2. In 7 h he will walk 2080 ⋅ 7 = 14560 m •

II –The fractional method or proportion. You must work as we have seen before with proportions. With the same data as in example:

5200 m 2,5 h 5200 m· 7 h = → x= =14650 m xm 7h 2,5 h It is a good idea to write always the units so we can see that we are organising the quantities correctly. 3. INVERSELY PROPORTIONAL MAGNITUDES We say that there is an inverse proportionality between two magnitudes if an increase in one magnitude causes a proportional decrease in the other and a decrease in the first magnitude causes a proportional increase in the other. That is if one magnitude is multiplied by 2, 3, ... this causes in the second a division by 2, 3, ... etc. If two values x and y are inversely proportional to each other then the product x·y is a constant. Team tasks are often an example of this. The time taken to do a job is indirectly proportional to the number of people in the team. Example: If 18 men can do a job in 10 days, in how many days will 45 men do the same job? This is an inverse proportion because with double the men, half the days are required. We will work in two steps. 1. Write the proportion (be careful! The same magnitude on each side)

18 men 10 days = 45 men x days 2. Make the inverse in one ratio

18 men x days = 45 men 10 days 3. Solve as we have done previously

x=

18· 10 = 4 days 45

4. PERCENTAGES Percentages often indicate proportions.

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Maths Department / I.E.S Al-qázeres

Numerical Proportionality

Mathematics- 4th Secondary Option A

(For example, labels in clothes indicate the various proportions of different yarns in thfabric. ‘Per cent’ means ‘per hundred’ and is denoted by the symbol %. 60% cotton means that 60/100 (or 0.6) of the fabric is cotton. 40% polyester means that 40/100 (or 0.4) is polyester.) Fractions and decimals can also be converted to percentages, by multiplying by 100%. So, for example, 0.17, 0.3 and 3/4 can be expressed as percentages as follows: 0.17 × 100% = 17%; 0.3 × 100% = 30%;

Example: Calculate the 35% of 28 =

35 ⋅ 28 = 0.35 · 28 = 9.8 100

Sometimes we are interested in calculating the total from the percent. Example: In the class 13 students didn’t do their homework; this was 52% of the class. How many students are in this class? We can say

52 ------------ 100 total 13 ------------ x total

x=

so

13· 100 =25 students 52

4.1. Percentage increase decrease To calculate a number increased or decreased in a percentage, we can use a formula. If a is the % increase, c the initial quantity and f the final quantity, then: f=

(1+ 100a )

⋅c

If a is the % decrease, c the initial quantity and f the final quantity, then: f=

(1− 100a )

⋅c

Example: The population of a town is 63500 and last year it increased by 8%, what is the population now? 63500 ⋅

(1+ 1008 )

=

63500 ·1.08 = 5080

Example: The price of some clothes is 68€ and there is a discount of 7%, what is the final price?

68 ⋅

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(1− 1007 )

=

68·( 1−0.07)=68 · 0.93 = 63.24 €

Maths Department / I.E.S Al-qázeres

Numerical Proportionality

Mathematics- 4th Secondary Option A

4.2. Finding the original amount If we know the % increase or decrease and the value we can find the total using proportions. Example:The net salary of an employee is 1230€ after paying 18% of IRPF, what is the gross of his/her salary? 82€ net --------- 100€ gross 1230€ net --------- x€ gross

x=

1230 ·100 =1500 € 82

Or using the percentage decrease:

100−18 1230 · x=1230→ (82 /100) · x=1230→ x= =1500 € 100 0.82 5. MIXTURES These exercises are about how to find the price of a mixture of several quantities of products with different prices in order not to have any profit or loss when they are sold or, in other occasions, having a certain benefit. Example: A mixture consists of 15 kg of coffee purchased at 8€/kg and chicory purchased at 2€/kg, how many kg of chicory do we need to mix if we want to sell the mixture at 6€/kg? We organize our calculations in a similar way : Item

difference of price

Nº of kg

Coffee

8 – 6 = 2€

15

Chicory

6 – 2 = 4€

x

total 30€= 15 · 2 loos 4x€

As the loss must be equal to the gain 4 x = 30 ⇒ x =

gain

30 = 7.5 kg 4

But you can do as well:

15· 8+ x · 2=(15+ x) · 6 120+2x=90+6x →120−90=6x−2x → 30=4x → x=

30 =7,5 kg 4

6. SIMPLE AND COMPOUND INTEREST When you deposit money in a bank, the bank usually pays you for the use of your money.

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If we consider that the annual interest is not added to the principal (original amount), this is SIMPLE INTEREST. In this case the formula for the interest or benefit after t years is:

Maths Department / I.E.S Al-qázeres

Numerical Proportionality

Mathematics- 4th Secondary Option A

I=

C ·r ·t 100

Where: I is the total interest, C is the principal, r is the % rate interest and t is the number of periods, usually years. The formula for the interest or benefit after t months is:

I=

C ·r ·t 1200

The formula for the interest or benefit after t days is:

I=

C·r ·t 36000

Example: Sarah deposits €4,000 at a bank at an interest rate of 4.5% per year. How much interest will she earn at the end of 3 years? Solution:

I=

4000 · 4,5 ·3 =540 € 100

She earns €540 at the end of 3 years.

•

If we consider that the annual interest is added to the principal (original amount) and interest in the next period will be generated on the now increased original amount, this is COMPOUND INTEREST. In this case the formula is:

(

C f =C · 1+

r 100

t

)

Where: C f is the amount after t years, C is the principal (original amount), r is the % rate interest and t is the number of years.

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Maths Department / I.E.S Al-qázeres

Numerical Proportionality

Mathematics- 4th Secondary Option A

7. SOLVED PROBLEMS •

A farmer needs 294 kg of fodder to feed 15 cows for a week. How many kg of fodder will he need to feed 10 cows 30 days?

Solution: COWS

DAYS

KILOGRAMS

15

7

294

15

1

294 : 7 = 42

1

1

42 : 15 = 2.8

10

1

2.8 · 10 = 28

10

30

28 · 30 = 840

To feed 10 cows during 30 days he will need 840 kg of fodder. •

It takes a group of men 15 days to build 400 squared metres of wall working 8 hours a day. How long will it take them to build 600 squared metres of wall if decide to work 10 hours a day?

Solution: SQUARED METRES

HOURS/DAY

DAYS

400

8

15

400

1

15 · 8 = 120

100

1

120 : 4 = 30

100

10

30 : 10 = 3

600

10

3 · 6 = 18

It will take them 18 days to build 600 squared metres working 10 hours/day. •

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Wanda borrowed €3,000 from a bank at an interest rate of 12% per year for a 2-year period. How much interest does she have to pay the bank at the end of 2 years?

Maths Department / I.E.S Al-qázeres

Numerical Proportionality

Mathematics- 4th Secondary Option A

Solution : Simple Interest = 3,000 × 12% × 2 = 720 She has to pay the bank €720 at the end of 2 years. •

Raymond bought a car for €40, 000. He took a $20,000 loan from a bank at an interest rate of 15% per year for a 3-year period. What is the total amount (interest and loan) that he would have to pay the bank at the end of 3 years?

Solution: Simple Interest = 20,000 × 13% × 3 = 7,800 At the end of 3 years, he would have to pay €20,000 + €7,800 = €27,800

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Maths Department / I.E.S Al-qázeres

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