- 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.

A \(\dfrac{3}{6}\)

B \(\dfrac{3}{9}\)

C \(\dfrac{3}{12}\)

D \(\dfrac{4}{20}\)

- 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.

A \(\dfrac{4}{6}\)

B \(\dfrac{6}{4}\)

C \(\dfrac{1}{3}\)

D \(\dfrac{2}{4}\)

**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.

- 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, \(\text{Scale Ratio}=\dfrac{\text{Measurement in drawing}}{\text{Actual Measurement}}\)

\(=\dfrac{\text{10 inches}}{\text{750 feet}}\)

\(=\dfrac{\text{1 inch}}{\text{75 feet}}\)

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 \(\dfrac{\text{10 inches}}{\text{1 feet}}\)

**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.

A \(20\)

B \(96\)

C \(16\)

D \(64\)

**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.

A \(9\; inches\)

B \(2\; inches\)

C \(1\; inch\)

D \(4\; inches\)

A \(1\;pound\)

B \(2\;pounds\)

C \(5\;pounds\)

D \(3\;pounds\)