Informative line

# Defining 'Means' and 'Extremes' of Proportion and the Condition of Proportion

A proportion represents two equivalent ratios.

For example:

$$\dfrac {3}{2}=\dfrac {6}{4}$$ or $$3:2=6:4$$ represents a proportion.

• If $$a:b=c:d$$ represents a proportion and $$a,\;b,\;c$$$$d$$ are the terms of the proportion, then
• 'Means' are $$b$$ and $$c\rightarrow$$ the closest terms
• 'Extremes' are $$a$$ and $$d\rightarrow$$ the furthest terms

## Condition of proportion

If the product of means equals to the product of extremes then the two ratios are in proportion.

So,

the product of means $$=c.b$$

Product of extremes $$=a.d$$

i.e.

That means $$a:b$$ and $$c:d$$ are in proportion.

#### Which one of the following ratios forms a proportion?

A $$5:3\text { and } 15:12$$

B $$1:2\text { and } 3:8$$

C $$4:5\text { and } 16:20$$

D $$3:7\text { and } 24:21$$

×

In option (A)

$$5:3\text { and } 15:12$$

Both the ratios can be written as:

$$\Rightarrow\dfrac {5}{3}=\dfrac {15}{12}$$

By applying the cross multiplication method, we get:

$$\Rightarrow3×15=5×12$$

$$\Rightarrow45\neq60$$

Here, the product of means is not equal to the product of extremes.

Thus, the ratios are not in proportion.

Hence, option (A) is incorrect.

In option (B)

$$1:2\text { and } 3:8$$

Both the ratios can be written as:

$$\Rightarrow\dfrac {1}{2}=\dfrac {3}{8}$$

By applying the cross multiplication method, we get:

$$\Rightarrow1×8=2×3$$

$$\Rightarrow8\neq6$$

Here, the product of means is not equal to the product of extremes.

Thus, the ratios are not in proportion.

Hence, option (B) is incorrect.

In option (C)

$$4:5\text { and } 16:20$$

Both the ratios can be written as:

$$\Rightarrow\dfrac {4}{5}=\dfrac {16}{20}$$

By applying the cross multiplication method, we get:

$$\Rightarrow4×20=5×16$$

$$\Rightarrow 80=80$$

Here, the product of means is equal to the product of extremes.

Thus, the ratios are in proportion.

Hence, option (C) is correct.

In option (D)

$$3:7\text { and } 24:21$$

Both the ratios can be written as:

$$\Rightarrow\dfrac {3}{7}=\dfrac {24}{21}$$

By applying the cross multiplication method, we get:

$$\Rightarrow3×21=7×24$$

$$\Rightarrow63\neq168$$

Here, the product of means is not equal to the product of extremes.

Thus, the ratios are not in proportion.

Hence, option (D) is incorrect.

### Which one of the following ratios forms a proportion?

A

$$5:3\text { and } 15:12$$

.

B

$$1:2\text { and } 3:8$$

C

$$4:5\text { and } 16:20$$

D

$$3:7\text { and } 24:21$$

Option C is Correct

# Finding Missing Term Using Proportion Reasoning

• A proportion shows that two ratios are equal.

For example: $$\dfrac {3}{2}=\dfrac {9}{6}$$ is a proportion.

• Two ratios which are in proportion have four terms which mean each ratio has two terms, as shown in the above example.
• Sometimes, we know only three terms out of the four.
• To solve this problem, assume the missing term as a variable.

For example: $$\dfrac {1}{3}=\dfrac {x}{9}$$

Here, the missing term is $$x$$.

To solve this proportion, follow the method given below:

Step 1: Find the relationship between either the numerators or the denominators.

Here, one of the numerators is missing, so we consider the denominators.

To get $$9$$ on the left side of the denominator, multiply it by $$3$$.

$$3×3=9$$

$$\therefore$$  The numerator also gets multiplied by $$3$$.

$$1×3=3$$

Step 2: Write the proportion again and find the value of the variable.

$$\dfrac {3}{9}=\dfrac {x}{9}$$

Thus, $$x=3$$

Step 3: Put the value of the variable in the given proportion.

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

The above method is called proportional reasoning.

#### Find the value of $$n$$ in the given proportion: $$\dfrac {17}{34}=\dfrac {n}{2}$$

A $$2$$

B $$3$$

C $$1$$

D $$0$$

×

Given proportion:

$$\dfrac {17}{34}=\dfrac {n}{2}$$

To get $$2$$ in the denominator, divide the denominator by $$17$$.

$$34\div17=2$$

$$\therefore$$ The numerator also gets divided by $$17$$.

$$17\div17=1$$

Now, write the proportion again and find the value of the variable.

$$\dfrac {1}{2}=\dfrac {n}{2}$$

Thus, $$n=1$$

Hence, option (C) is correct.

### Find the value of $$n$$ in the given proportion: $$\dfrac {17}{34}=\dfrac {n}{2}$$

A

$$2$$

.

B

$$3$$

C

$$1$$

D

$$0$$

Option C is Correct

# 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

# 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

# Finding the  Missing Term Using Cross Products

To understand it in a better way, consider an example.

For example: Jessica wants to make almond butter cookies. The recipe calls $$6$$ cups of flour for every $$4$$ cups of sugar. If she wants to make the cookies using $$9$$ cups of flour, how much sugar does she need?

We can solve the above problem by using proportion.

Step 1: Assume the asked quantity as a variable.

Let she uses $$x$$ cups of sugar for $$9$$ cups of flour.

Step 2: Write the proportion.

$$\dfrac {6 \text { cups of flour}}{4 \text{ cups of sugar}}= \dfrac {9 \text { cups of flour}}{x \text{ cups of sugar}}$$

$$\dfrac {6}{4}= \dfrac {9}{x}$$

Step 3: Solve the problem by using cross product.

$$6×x=9×4$$

$$6x=36$$

Divide both the sides by $$6$$.

$$\dfrac {6x}{6}=\dfrac {36}{6}$$

$$x=6$$

Thus, she should use $$6$$ cups of sugar.

#### Julie spends $$16$$ hours in $$2$$ weeks period practicing her culinary skills. How many hours does she practice in $$5$$ weeks?

A $$40$$ hours

B $$30$$ hours

C $$50$$ hours

D $$45$$ hours

×

Julie spends $$16$$ hours in $$2$$ weeks to practice her culinary skills.

Let she spends $$x$$ hours in $$5$$ weeks.

Write the proportion.

$$\dfrac {16\text { hours}} {2\text { weeks}}= \dfrac {x\text { hours}} {5\text { weeks}}$$

$$\Rightarrow\dfrac {16}{2}=\dfrac {x}{5}$$

Apply the cross multiplication method.

$$\Rightarrow 2×x=16×5$$

$$\Rightarrow 2x=80$$

Divide both the sides by $$2$$.

$$\dfrac {2x}{2}=\dfrac {80}{2}$$

$$x=40$$

Thus, she practices for $$40$$ hours in $$5$$ weeks.

Hence, option(A) is correct.

### Julie spends $$16$$ hours in $$2$$ weeks period practicing her culinary skills. How many hours does she practice in $$5$$ weeks?

A

$$40$$ hours

.

B

$$30$$ hours

C

$$50$$ hours

D

$$45$$ hours

Option A is Correct