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Overview

Proportions are a method of relating one quantity to another.

Proportional reasoning is the ability to see that two quantities change in a fixed ratio. On the AMC 8 this appears in recipe scaling, map measurements, similar figures, and rate problems.

Master these ideas and you will unlock shortcuts throughout algebra and geometry.

Fractions (Skip this if you already know)

Fractions are a way of representing parts of a whole. A fraction tells us how many parts we have out of the total number of equal parts. A fraction is written in the form:

ab\frac{a}{b}

where:

  • (a) is called the numerator, it tells us how many parts we have.
  • (b) is called the denominator, it tells us how many equal parts the whole is divided into.

For example:

34\frac{3}{4}

means that a whole is divided into 4 equal parts, and we take 3 of those parts.

An example from daily life: Suppose a pizza is cut into 8 equal slices. If you eat 3 slices, then you have eaten:

38\frac{3}{8}

of the pizza.

Operations Using Fractions

Just like whole numbers, fractions can also be added, subtracted, multiplied, and divided.

Addition of Fractions

If the denominators are the same, we simply add the numerators.

For example:

25+15=35\frac{2}{5} + \frac{1}{5} = \frac{3}{5}

This means that if we take 2 parts out of 5 and then another 1 part out of 5, we now have 3 parts out of 5.

If the denominators are different, we first convert them into equivalent fractions with the same denominator.

For example:

12+14\frac{1}{2} + \frac{1}{4}

We convert 12 \frac{1}{2}\ into 24\frac{2}{4} :

24+14=34\frac{2}{4} + \frac{1}{4} = \frac{3}{4}

Subtraction of Fractions

Fractions are subtracted in a similar way.

For example:

56−26=36=12\frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}

This means that if we remove 2 parts out of 6 from 5 parts out of 6, we are left with 3 parts out of 6.

Multiplication of Fractions

To multiply fractions, we multiply the numerators together and the denominators together.

For example:

23×45=815\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}

This operation is useful when finding a fraction of another fraction.

For example, finding half of three-fourths:

12×34=38\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}

Division of Fractions

To divide fractions, we multiply by the reciprocal of the second fraction.

For example:

34÷25\frac{3}{4} \div \frac{2}{5}

becomes:

34×52=158\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}

This tells us how many times one fraction fits into another.

Fractions and Proportions

Fractions and proportions are closely related because both compare quantities.

A fraction compares a part to a whole, while a proportion compares two ratios or fractions.

For example:

24=12\frac{2}{4} = \frac{1}{2}

Both fractions represent the same proportion because they describe the same relationship between the numerator and denominator.

An example from daily life: Suppose 2 out of every 5 students in a class wear glasses. Then the fraction:

25\frac{2}{5}

represents the proportion of students wearing glasses.

If another class has 4 students wearing glasses out of 10 students:

410\frac{4}{10}

then:

25=410\frac{2}{5} = \frac{4}{10}

This means that both classes have the same proportion of students wearing glasses.

Cross Multiplication

Cross multiplication is a method used when two fractions are equal. It helps us find an unknown value in a proportion.

Suppose:

ab=cd\frac{a}{b} = \frac{c}{d}

Then, by cross multiplication:

a×d=b×ca \times d = b \times c

This works because the cross products of equal fractions are always equal.

For example:

x+2yx−y=34\frac{x + 2y}{x - y} = \frac{3}{4}

Cross multiplying gives:

4(x+2y)=3(x−y)4(x + 2y) = 3(x - y)

Expanding both sides:

4x+8y=3x−3y4x + 8y = 3x - 3y

Bringing like terms together:

x=−11yx = -11y

Cross multiplication is especially useful when solving algebraic equations involving ratios and proportions.

Direct and Inverse Proportions

Proportions are a method of relating one quantity to another. If one of the quantities increases, the other does too, then it's called a Direct Proportion. If one of the quantities increases, the other decreases, then it's called a Inverse Proportion.

An example of Direct Proportion: The cost of apples is 2 dollars per kg, then for each apple you buy, you pay an additional 2 dollars. This means that the amount you pay increases with the no of apples you buy. This means that the amount you pay and the number of apples you get are in Direct Proportion.

An example of Inverse Proportion: The time taken to travel a fixed distance is 120 km. If you travel faster, you take less time, and if you travel slower, you take more time. This means that as speed increases, time decreases in such a way that their product remains constant for the same distance. This means that the speed and the time taken to cover the distance are in inverse proportion.

Direct Propotions are represented as

y∝x or y=kx\boxed{y \propto x } \text{ or } \boxed{y = kx}

Inverse Proportions are represented as

y∝1x or y=kx or xy=k\boxed{y \propto \frac{1}{x}} \text{ or } \boxed{y = \frac{k}{x}} \text{ or } \boxed{xy = k}

Note: kk is a propotionality constant and can be any number.

Percentages

Percentages are a way of expressing a quantity out of 100. The word percentage means “per hundred”.

The symbol used for percentage is:

%\%

For example:

50%50\%

means:

50100=12\frac{50}{100} = \frac{1}{2}

This means that 50% represents half of a whole.

An example from daily life: If a student scores 45 marks out of 50, then the percentage score is:

4550×100=90%\frac{45}{50} \times 100 = 90\%

This means the student scored 90 percent of the total marks.

Converting Between Fractions, Decimals, and Percentages

Fractions, decimals, and percentages are different ways of representing the same value.

Fraction to Percentage

To convert a fraction into a percentage, multiply by 100.

For example:

34×100=75%\frac{3}{4} \times 100 = 75\%

Percentage to Fraction

To convert a percentage into a fraction, divide by 100.

For example:

25%=25100=1425\% = \frac{25}{100} = \frac{1}{4}

Decimal to Percentage

To convert a decimal into a percentage, multiply by 100.

For example:

0.6×100=60%0.6 \times 100 = 60\%

Percentage to Decimal

To convert a percentage into a decimal, divide by 100.

For example:

80%=0.880\% = 0.8

Increase and Decrease in Percentages

Percentages are often used to describe increases or decreases in quantities.

Percentage Increase

If the price of a book increases from 200 dollars to 240 dollars, then the increase is:

240−200=40240 - 200 = 40

The percentage increase is:

40200×100=20%\frac{40}{200} \times 100 = 20\%

Percentage Decrease

If the price decreases from 200 dollars to 150 dollars, then the decrease is:

200−150=50200 - 150 = 50

The percentage decrease is:

50200×100=25%\frac{50}{200} \times 100 = 25\%

Scaling

Scaling means enlarging or reducing a quantity while keeping the same proportion.

When something is scaled, all corresponding lengths or quantities change by the same factor.

For example, suppose a map uses the scale:

1 cm:10 km1 \text{ cm} : 10 \text{ km}

This means that every 1 cm on the map represents 10 km in real life.

If two cities are 5 cm apart on the map, then the actual distance is:

5×10=50 km5 \times 10 = 50 \text{ km}

Scale Factor

The number by which quantities are multiplied is called the scale factor.

For example, if a square of side length 2 cm is enlarged to a square of side length 6 cm, then:

Scale Factor=62=3\text{Scale Factor} = \frac{6}{2} = 3

This means every side becomes 3 times larger.

Conversion Factors

Conversion factors are fractions equal to (1) that help us change units.

For example:

1 inch=2.54 cm1 \text{ inch} = 2.54 \text{ cm}

So we can write:

1 inch2.54 cmor2.54 cm1 inch\frac{1 \text{ inch}}{2.54 \text{ cm}} \quad \text{or} \quad \frac{2.54 \text{ cm}}{1 \text{ inch}}

Both are equal to (1).

To convert (180) centimeters into inches:

180 cm×1 inch2.54 cm≈70.9 inches180 \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} \approx 70.9 \text{ inches}

The centimeters cancel, leaving inches.

We choose the conversion factor so that unwanted units cancel out.

Squared and Cubed Units

When converting areas or volumes, the conversion factor must also be squared or cubed.

For example:

1 ft=0.3048 m1 \text{ ft} = 0.3048 \text{ m}

Then:

1 ft2=(0.3048)2 m21 \text{ ft}^2 = (0.3048)^2 \text{ m}^2

So:

5 ft2×(0.3048 m1 ft)2≈0.4645 m25 \text{ ft}^2 \times \left( \frac{0.3048 \text{ m}}{1 \text{ ft}} \right)^2 \approx 0.4645 \text{ m}^2

The same idea applies for cubic units.

Negative Powers in Units

Sometimes units are written using negative powers.

For example:

m−1=1m\text{m}^{-1} = \frac{1}{\text{m}}

and:

g cm−3=gcm3\text{g cm}^{-3} = \frac{\text{g}}{\text{cm}^3}

Negative powers simply indicate that the unit appears in the denominator.

For example, speed can be written as:

m s−1=ms\text{m s}^{-1} = \frac{\text{m}}{\text{s}}

Similarly, density is often written as:

kg m−3=kgm3\text{kg m}^{-3} = \frac{\text{kg}}{\text{m}^3}

Example

Convert:

72 km h−172 \text{ km h}^{-1}

into meters per second.

Using:

1 km=1000 m1 \text{ km} = 1000 \text{ m}

and:

1 h=3600 s1 \text{ h} = 3600 \text{ s}

we write:

72 km h−1×1000 m1 km×1 h3600 s72 \text{ km h}^{-1} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}}

The km and h units cancel, leaving:

72×10003600=2072 \times \frac{1000}{3600} = 20

Thus:

72 km h−1=20 m s−172 \text{ km h}^{-1} = 20 \text{ m s}^{-1}

When converting such units, we still use conversion factors normally, making sure the units cancel correctly.

Part 2

Great work if you read all the way till here! Due to File-Size Constraints, this module has been divided into 2 parts. Continue this Module in part 2. Good Luck!

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