Informative line

# Proportion

• Proportions are everywhere around us. These are the comparisons that we make among different things.

Proportion

• If two ratios are equal, we say that they are in proportion.
• It is denoted by the symbol  '$$::$$' or '$$=$$'.
• A ratio compares two quantities and a proportion compares two equal ratios.
• Two ratios are in proportion if their simplest forms are same.

For example: $$\dfrac{1}{4}$$ and $$\dfrac{5}{20}$$

Do they form a proportion?

Since $$\dfrac{1}{4}$$ is in its simplest form, thus, we leave that one as it is.

Now, consider $$\dfrac{5}{20}$$

The GCF of $$5$$ and $$20$$ is $$5$$.
$$\dfrac{5\div5}{20\div5}=\dfrac{1}{4}$$

As $$\dfrac{1}{4}=\dfrac{1}{4}$$

Thus, $$\dfrac{1}{4}$$ and $$\dfrac{5}{20}$$ form a proportion.

#### Which one of the following forms a proportion with $$\dfrac{2}{8}$$?

A $$\dfrac{3}{6}$$

B $$\dfrac{3}{9}$$

C $$\dfrac{3}{12}$$

D $$\dfrac{4}{20}$$

×

Two ratios are in proportion if their simplest forms are same.

The given ratio is $$\dfrac{2}{8}$$.

The GCF of $$2$$ and $$8$$ is $$2$$.

$$\dfrac{2\div2}{8\div2}=\dfrac{1}{4}$$

Thus, the simplest form of $$\dfrac{2}{8}$$ is $$\dfrac{1}{4}$$.

Option (A) is $$\dfrac{3}{6}$$.

The GCF of $$3$$ and $$6$$ is $$3$$.

$$\dfrac{3\div3}{6\div3}=\dfrac{1}{2}$$

As $$\dfrac{1}{4}\ne\dfrac{1}{2}$$

Thus, option (A) is incorrect.

Option (B) is $$\dfrac{3}{9}$$.

The GCF of $$3$$ and $$9$$ is $$3$$.

$$\dfrac{3\div3}{9\div3}=\dfrac{1}{3}$$

As $$\dfrac{1}{4}\ne\dfrac{1}{3}$$

Thus, option (B) is incorrect.

Option (C) is $$\dfrac{3}{12}$$.

The GCF of $$3$$ and $$12$$ is $$3$$.

$$\dfrac{3\div3}{12\div3}=\dfrac{1}{4}$$

As $$\dfrac{1}{4}=\dfrac{1}{4}$$

Thus, option (C) is correct.

Option (D) is $$\dfrac{4}{20}$$.

The GCF of $$4$$ and $$20$$ is $$4$$.

$$\dfrac{4\div4}{20\div4}=\dfrac{1}{5}$$

As $$\dfrac{1}{4}\ne\dfrac{1}{5}$$

Thus, option (D) is incorrect.

### Which one of the following forms a proportion with $$\dfrac{2}{8}$$?

A

$$\dfrac{3}{6}$$

.

B

$$\dfrac{3}{9}$$

C

$$\dfrac{3}{12}$$

D

$$\dfrac{4}{20}$$

Option C is Correct

# Cross Multiplication or Cross Product

• By cross multiplication or cross product, we can check whether the two ratios form a proportion or not.

Cross Product

• A cross product is, when we multiply the numerator of one ratio with the denominator of another.
• If the products are same, they form a proportion otherwise not.

For example: $$\dfrac{4}{3}\;and \;\dfrac{5}{6}$$

We will multiply $$4$$ by $$6$$ and $$3$$ by $$5$$

$$4\times6=24$$ and $$3\times5=15$$

As $$24\ne15$$

Thus, $$\dfrac{4}{3}\text{ and }\dfrac{5}{6}$$ do not form a proportion.

We will multiply $$4$$ by $$6$$ and $$3$$ by $$5$$

$$4\times6=24$$

and $$3\times5=15$$

Since $$24\ne15$$

Thus, $$\dfrac{4}{3}\text{ and }\dfrac{5}{6}$$ do not form a proportion.

#### Which one of the following forms a proportion with $$\dfrac{2}{3}$$? (Use cross multiplication)

A $$\dfrac{4}{6}$$

B $$\dfrac{6}{4}$$

C $$\dfrac{1}{3}$$

D $$\dfrac{2}{4}$$

×

Using cross product, we can check whether the two ratios form a proportion or not.

For option (A),

$$\dfrac{2}{3}$$ and $$\dfrac{4}{6}$$

$$2\times6=12$$ and $$3\times4=12$$

As $$12=12$$

Thus, $$\dfrac{2}{3}$$ and $$\dfrac{4}{6}$$ form a proportion.

Hence, option (A) is correct.

For option (B),

$$\dfrac{2}{3}$$ and $$\dfrac{6}{4}$$

$$2\times4=8$$ and $$3\times6=18$$

As $$8\ne18$$

Thus, $$\dfrac{2}{3}$$ and $$\dfrac{6}{4}$$ do not form a proportion.

Hence, option (B) is incorrect.

For option (C),

$$\dfrac{2}{3}$$ and $$\dfrac{1}{3}$$

$$2\times3=6$$ and $$3\times1=3$$

As $$6\ne3$$

Thus, $$\dfrac{2}{3}$$ and $$\dfrac{1}{3}$$ do not form a proportion.

Hence, option (C) is incorrect.

For option (D),

$$\dfrac{2}{3}$$ and $$\dfrac{2}{4}$$

$$2\times4=8$$ and $$3\times2=6$$

As $$8\ne6$$

Thus, $$\dfrac{2}{3}$$ and $$\dfrac{2}{4}$$ do not form a proportion.

Hence, option (D) is incorrect.

### Which one of the following forms a proportion with $$\dfrac{2}{3}$$? (Use cross multiplication)

A

$$\dfrac{4}{6}$$

.

B

$$\dfrac{6}{4}$$

C

$$\dfrac{1}{3}$$

D

$$\dfrac{2}{4}$$

Option A is Correct

# Proportion through Figures

Proportion

• A proportion shows that two ratios are equal.

For example: Consider the blocks as shown in the figure. We will check whether the ratios of shaded parts of both the blocks are in proportion.

In Block I,

Number of shaded parts $$=2$$

Total number of parts $$=6$$

Thus, the ratio becomes $$2:6=\dfrac{2}{6}$$

In Block II,

Number of shaded parts $$=3$$

Total number of parts $$=9$$

Thus, the ratio becomes $$3:9=\dfrac{3}{9}$$

Now, if the ratios of shaded parts of both the blocks are in proportion, the ratios must be equal.

$$\Rightarrow\dfrac{2}{6}=\dfrac{3}{9}$$

By cross product,

$$2\times9=18$$

$$6\times3=18$$

As $$18=18$$

Thus, shaded parts of both the blocks are in proportion.

$$2\times9=18$$

$$6\times3=18$$

$$\because \; 18=18$$

Thus, both blocks are in proportion.

#### Consider the blocks as shown in the figure. Which option shows the correct number of shaded parts of Block II, when the ratios of shaded parts of both the blocks are in proportion?

A $$5$$

B $$6$$

C $$8$$

D $$9$$

×

In Block I,

Number of shaded parts $$=10$$

Total number of parts $$=20$$

Thus, the ratio becomes $$10:20=\dfrac{1}{2}$$

In Block II,

Number of shaded parts $$=x$$

Total number of parts $$=12$$

Thus, the ratio becomes $$x:12=\dfrac{x}{12}$$

Now, if shaded parts of both the blocks are in proportion, the ratios must be equal.

$$\dfrac{1}{2}=\dfrac{x}{12}$$

So, by cross product,

$$\Rightarrow1\times12=2x$$

$$\Rightarrow12=2x$$

By using the inverse operation of multiplication, divide by $$2$$ on both the sides.

$$\Rightarrow\dfrac{12}{2}=\dfrac{2x}{2}$$

$$\Rightarrow x=\dfrac{12}{2}$$

$$\Rightarrow x=6$$

This means the shaded portion of Block II is $$6$$ parts.

Hence, option (B) is correct.

### Consider the blocks as shown in the figure. Which option shows the correct number of shaded parts of Block II, when the ratios of shaded parts of both the blocks are in proportion?

A

$$5$$

.

B

$$6$$

C

$$8$$

D

$$9$$

Option B is Correct

# Proportions to Find Scale Ratios

• Here, we will learn to find the dimensions through proportions.
• First, we need to know about the scale drawing.

Scale drawing

• A scale drawing is a drawing that is used to represent any object on a sheet of paper which is very large to be drawn in its actual dimensions.

For example: A $$27$$ feet tall building can be $$3$$ inches tall on a sheet of paper.

Scale Ratio / Scale Factor

• Scale ratio is the ratio of measurement in drawing to the measurement of actual object.

For example: A lane in a residential block is $$750$$ feet long. It is $$10$$ inches long in the drawing.

What is the scale ratio?

Here, actual measurement $$=750$$ feet

Measurement in drawing $$=10$$ inches

Thus, $$\text{Scale Ratio}=\dfrac{\text{Measurement in drawing}}{\text{Actual Measurement}}$$

$$=\dfrac{\text{10 inches}}{\text{750 feet}}$$

$$=\dfrac{\text{1 inch}}{\text{75 feet}}$$

#### The actual height of a building is $$150$$ feet. In drawing, it is shown as $$3$$ inches. What is the scale ratio?

A $$\dfrac{\text{50 feet}}{\text{1 inch}}$$

B $$\dfrac{\text{1 inch}}{\text{10 feet}}$$

C $$\dfrac{\text{1 inch}}{\text{50 feet}}$$

D $$\dfrac{\text{10 inches}}{\text{1 feet}}$$

×

Given:

Actual measurement $$=150$$ feet

Measurement in drawing $$=3$$ inches

$$\text{Scale Ratio}=\dfrac{\text{Measurement in drawing}}{\text{Actual Measurement}}$$

$$=\dfrac{\text{3 inches}}{\text{150 feet}}$$

$$=\dfrac{\text{1 inch}}{\text{50 feet}}$$

Hence, option (C) is correct.

### The actual height of a building is $$150$$ feet. In drawing, it is shown as $$3$$ inches. What is the scale ratio?

A

$$\dfrac{\text{50 feet}}{\text{1 inch}}$$

.

B

$$\dfrac{\text{1 inch}}{\text{10 feet}}$$

C

$$\dfrac{\text{1 inch}}{\text{50 feet}}$$

D

$$\dfrac{\text{10 inches}}{\text{1 feet}}$$

Option C is Correct

# Finding Unknown Number in a Proportion

Proportion

• A proportion shows that two ratios are equal.
• We can find an unknown term (number) in a proportion through cross multiplication.

For example: What is the value of $$x$$ in the proportion below?

$$\dfrac{30}{48}=\dfrac{5}{x}$$

In the given problem, we will find $$x$$ by cross multiplication.

Since both the ratios are in proportion, thus, their cross products should be equal.

So, $$30x=48\times5$$

Dividing both sides by $$30$$,

$$\dfrac{30x}{30}=\dfrac{48\times5}{30}$$

$$x=\dfrac{48}{6}$$

$$x=8$$

Thus, $$x=8$$ holds the proportion to be true.

Since both the ratios are in proportion, thus, their cross products should be equal.

Hence, $$30x=48\times5$$

Dividing both the sides by $$30$$,

$$\dfrac{30x}{30}=\dfrac{48\times5}{30}$$

$$x=\dfrac{48}{6}$$

$$x=8$$

Thus, $$x=8$$ holds the proportion to be true.

#### Which value of $$x$$ makes the given proportion true? $$\dfrac{4}{24}=\dfrac{16}{x}$$

A $$20$$

B $$96$$

C $$16$$

D $$64$$

×

Given: $$\dfrac{4}{24}=\dfrac{16}{x}$$

We will find the value of $$x$$ by cross multiplication.

Since both the ratios are in proportion, thus, their cross products are equal.

$$4x=24\times16$$

Dividing both sides by $$4$$,

$$\dfrac{4x}{4}=\dfrac{24\times16}{4}$$

$$x=6\times16$$

$$x=96$$

Hence, option (B) is correct.

### Which value of $$x$$ makes the given proportion true? $$\dfrac{4}{24}=\dfrac{16}{x}$$

A

$$20$$

.

B

$$96$$

C

$$16$$

D

$$64$$

Option B is Correct

# Proportion to Find Dimensions

Proportion

• A proportion shows that two ratios are equal.
• We can find dimensions of an object through proportions (using cross product).

For example: The scale ratio of a building is $$\text{1 inch : 7 m}$$. The actual height of the building is $$\text{28 m}$$. How tall is the building in the drawing?

Here, we have to calculate the height of the building in drawing.

Scale Ratio $$=\dfrac{\text{1 inch}}{\text{7 m}}$$

Actual height $$=\text{28 m}$$

Let the height of the building in drawing be $$x$$ inches.

So, the ratio is  $$\dfrac{x\text{ inches}}{\text{28 m}}$$

Both the ratios will form a proportion.

So, $$\dfrac{1}{7}=\dfrac{x}{28}$$

By cross product,

$$7x=28$$

Divide both sides by $$7$$,

$$\Rightarrow\dfrac{7x}{7}=\dfrac{28}{7}$$

$$\Rightarrow x=4$$

Hence, the building is $$4$$ inches tall in the drawing.

$$7x=28$$

Dividing both the sides by $$7$$

$$\Rightarrow\dfrac{7x}{7}=\dfrac{28}{7}$$

$$\Rightarrow x=4$$

Hence, building is $$4$$ inches long in the drawing.

#### The scale ratio of the length of a park is $$\text{1 inch : 10 m}$$. The actual length of the park is $$90$$ m. How long is the park in the drawing?

A $$9\; inches$$

B $$2\; inches$$

C $$1\; inch$$

D $$4\; inches$$

×

Given:

Scale Ratio $$=\dfrac{\text{1 inch}}{\text{10 m}}$$

Actual length $$=90 \;m$$

Let the length of the park in drawing be $$x$$ inches.

So, the ratio is  $$\dfrac{x \; inch}{90\;m}$$

Both the ratios form a proportion.

$$\dfrac{1}{10}=\dfrac{x}{90}$$

By cross product,

$$\Rightarrow10x=90$$

Divide both sides by $$10$$,

$$\Rightarrow \dfrac{10x}{10}=\dfrac{90}{10}$$

$$\Rightarrow x=9\; inches$$

Hence, the park is $$9$$ inches long in the drawing.

Hence, option (A) is correct.

### The scale ratio of the length of a park is $$\text{1 inch : 10 m}$$. The actual length of the park is $$90$$ m. How long is the park in the drawing?

A

$$9\; inches$$

.

B

$$2\; inches$$

C

$$1\; inch$$

D

$$4\; inches$$

Option A is Correct

#### Sarah bought $$4$$ pounds of apples for $$24$$. How many pounds of apples can Sarah buy if she has $$18$$?

A $$1\;pound$$

B $$2\;pounds$$

C $$5\;pounds$$

D $$3\;pounds$$

×

Given ratio $$=\dfrac{4\;pounds}{24}$$

Let the quantity of apples Sarah can buy for $$18$$ be $$x$$.

Thus, the ratio becomes $$\dfrac{x}{18}$$.

Both the ratios form a proportion.

$$\dfrac{4}{24}=\dfrac{x}{18}$$

By cross product,

$$\Rightarrow18\times4=24x$$

Dividing both sides by $$24$$,

$$\Rightarrow\dfrac{72}{24}=\dfrac{24x}{24}$$

$$\Rightarrow3=x$$

Thus, Sarah can buy $$3$$ pounds of apples for $$18$$.

Hence, option (D) is correct.

### Sarah bought $$4$$ pounds of apples for $$24$$. How many pounds of apples can Sarah buy if she has $$18$$?

A

$$1\;pound$$

.

B

$$2\;pounds$$

C

$$5\;pounds$$

D

$$3\;pounds$$

Option D is Correct