Informative line

# Simplest Form of a Ratio

• A ratio is in its simplest form if the numerator and the denominator do not have any common factor other than $$1.$$
• The simplest form of a ratio is also known as reduced form.

### How to simplify a ratio

• We simplify a ratio written in fraction form in the same way as we simplify a fraction.
• To simplify a ratio, find out the greatest common factor of numerator and denominator.
• Divide both numerator and denominator by the greatest common factor.
• We get the simplest form of ratio.

For example: $$\dfrac{2}{22}$$

Here, the GCF of  $$2$$ and $$22$$ is $$2.$$ Thus, we will divide both numerator and denominator by $$2.$$

$$\dfrac{2\div2}{22\div2}=\dfrac{1}{11}$$

Hence, the simplest form of $$\dfrac{2}{22}$$ is $$\dfrac{1}{11}$$.

Note: The values of a ratio and of simplified form are always same.

#### Which one of the following is the simplest form of $$\dfrac{12}{8}$$?

A $$\dfrac{1}{8}$$

B $$\dfrac{2}{3}$$

C $$\dfrac{3}{2}$$

D $$\dfrac{12}{1}$$

×

The greatest common factor of $$12$$ and $$8$$ is $$4.$$

So, divide both numerator and denominator by $$4.$$

$$\dfrac{12\div4}{8\div4}=\dfrac{3}{2}$$

Thus, the simplest form of $$\dfrac{12}{8}$$ is $$\dfrac{3}{2}$$.

Hence, option (C) is correct.

### Which one of the following is the simplest form of $$\dfrac{12}{8}$$?

A

$$\dfrac{1}{8}$$

.

B

$$\dfrac{2}{3}$$

C

$$\dfrac{3}{2}$$

D

$$\dfrac{12}{1}$$

Option C is Correct

# Equivalent Ratios

• The ratios which have the same simplified form are called equivalent ratios.

For example: $$\dfrac{1}{2}$$ and $$\dfrac{3}{6}$$

If we simplify $$\dfrac{3}{6}$$, it becomes $$\dfrac{1}{2}$$.

Thus, $$\dfrac{1}{2}$$ and $$\dfrac{3}{6}$$ are equivalent ratios.

• Equivalent ratios of a ratio can be obtained by two methods:
1. Scaling up (multiplication)
2. Scaling down (division)
• Scaling up (Multiplication):

The equivalent ratios of a ratio can be obtained by multiplying both numerator and denominator by a non-zero number.

For example: $$\dfrac{1}{3}$$

To obtain the equivalent ratio of $$\dfrac{1}{3}$$, multiply both $$1$$ and $$3$$ by a non-zero number.

Let it be $$3.$$

Then, $$\dfrac{1×3}{3×3}=\dfrac{3}{9}$$

Thus, the equivalent ratio of $$\dfrac{1}{3}$$ is $$\dfrac{3}{9}$$.

• Scaling down (Division):

The equivalent ratios of a ratio can be obtained by dividing both numerator and denominator by a non-zero number.

For example: $$\dfrac{22}{6}$$

To obtain the equivalent ratio of $$\dfrac{22}{6}$$, divide both $$22$$ and $$6$$ by a non-zero number.

The GCF of  $$22$$ and $$6$$ is $$2.$$

Thus, $$\dfrac{22\div2}{6\div2}=\dfrac{11}{3}$$

Thus, the equivalent ratio of $$\dfrac{22}{6}$$ is $$\dfrac{11}{3}$$.

#### Which one of the following options DOES NOT represent the equivalent ratio of $$\dfrac{2}{6}$$?

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

B $$\dfrac{4}{12}$$

C $$\dfrac{1}{6}$$

D $$\dfrac{12}{36}$$

×

First, we simplify the given ratio, i.e. $$\dfrac{2}{6}$$.

The GCF of  $$2$$ and $$6$$ is $$2.$$

Thus, $$\dfrac{2\div2}{6\div2}=\dfrac{1}{3}$$

Option (A): $$\dfrac{1}{3}$$

$$\dfrac{1}{3}$$ is the simplest form of the given ratio.

Thus, option (A) is incorrect.

Option (B): $$\dfrac{4}{12}$$

The GCF of $$4$$ and $$12$$ is $$4.$$

Thus, $$\dfrac{4\div4}{12\div4}=\dfrac{1}{3}$$

Hence, option (B) is incorrect.

Option (C): $$\dfrac{1}{6}$$

The GCF of $$1$$ and $$6$$ is $$1.$$

Thus, it is in its simplest form, which is not equal to $$\dfrac{1}{3}$$.

Thus, option (C) is correct.

Option (D): $$\dfrac{12}{36}$$

The GCF of $$12$$ and $$36$$ is $$12$$.

Thus, $$\dfrac{12\div12}{36\div12}=\dfrac{1}{3}$$

Hence, option (D) is incorrect.

### Which one of the following options DOES NOT represent the equivalent ratio of $$\dfrac{2}{6}$$?

A

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

.

B

$$\dfrac{4}{12}$$

C

$$\dfrac{1}{6}$$

D

$$\dfrac{12}{36}$$

Option C is Correct

# Ratio as Decimal

• A ratio can be written as a fraction and as a decimal.
• To convert a ratio to a decimal, we need to recall how to convert a fraction to a decimal.
• To convert a fraction to a decimal, divide the numerator by the denominator using long division method.
• Ratios can be converted to decimals in the following way:

Step 1: Write the ratio in fraction form.

Step 2: Simplify the fraction, if possible.

Step 3: Convert the fraction to a decimal.

For example: We want to convert $$3:6$$ in decimal form.

Step 1: We write the given ratio, $$3:6$$ in fraction form.

$$3:6=\dfrac{3}{6}$$

Step 2: Now, we have to simplify the fraction, if possible.

The GCF of $$3$$ and $$6$$ is $$3$$.

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

Step 3: We will convert the fraction to a decimal by dividing $$1$$ by $$2$$.

Thus, $$3:6=0.5$$  #### Which one of the following options represents the decimal form of $$4:5$$ ?

A $$0.5$$

B $$0.6$$

C $$0.7$$

D $$0.8$$

×

Given ratio: $$4:5$$

We write the given ratio $$4:5$$  in fraction form.

$$4:5$$  $$=\dfrac{4}{5}$$

Now, we simplify the fraction, if possible.

The GCF of $$4$$ and $$5$$ is $$1$$.

Thus, $$\dfrac{4}{5}$$ is already in its simplest form.

We convert the fraction to a decimal by dividing $$4$$ by $$5$$. Thus, $$\dfrac{4}{5}=0.8$$

Hence, option (D) is correct.

### Which one of the following options represents the decimal form of $$4:5$$ ?

A

$$0.5$$

.

B

$$0.6$$

C

$$0.7$$

D

$$0.8$$

Option D is Correct

# Comparison of Ratios

• Ratios can be compared by converting them to decimals.
• They can be converted to decimals in the following way:

Step 1: Write the ratios in fraction form.

Step 2: Simplify the fractions, if possible.

Step 3: Convert the fractions to decimals.

For example: $$2:4$$ and $$4:5$$ are to be compared.

Step 1: We write the given ratios,  $$2:4$$ and $$4:5$$ in fraction form.

$$2:4=\dfrac{2}{4}$$ and $$4:5=\dfrac{4}{5}$$

Step 2: Now, we simplify the fractions, if possible.

The GCF of $$2$$ and $$4$$ is $$2$$.

Thus, $$\dfrac{2\div2}{4\div2}=\dfrac{1}{2}$$

The GCF of $$4$$ and $$5$$ is $$1$$.

Thus, $$\dfrac{4}{5}$$ is already in its simplest form.

Step 3: We convert the fractions to decimals by dividing $$1$$ by $$2$$ and $$4$$ by $$5$$.

Thus, $$\dfrac{1}{2}=0.5$$ and $$\dfrac{4}{5}=0.8$$

Now, compare both the ratios.

$$0.5<0.8$$

Thus, $$\dfrac{2}{4}<\dfrac{4}{5}$$

or $$2:4<4:5$$.  #### Which one of the following ratios is greater,  $$\text{2 to 4 or 1 to 5}$$ ?

A $$\text{2 to 4}$$

B $$\text{1 to 5}$$

C $$\text{Both are equal}$$

D

×

Given ratios : $$\text{2 to 4 and 1 to 5}$$

We write the given ratios, $$\text{2 to 4}$$ and  $$\text{1 to 5}$$ in fraction form.

$$\text{2 to 4} =\dfrac{2}{4}$$ and $$\text{1 to 5}=\dfrac{1}{5}$$

Now, we simplify the fractions, if possible.

The GCF of $$2$$ and $$4$$ is $$2$$.

Thus, $$\dfrac{2\div2}{4\div2}=\dfrac{1}{2}$$

The GCF of $$1$$ and $$5$$ is $$1$$.

Thus, $$\dfrac{1}{5}$$ is already in its simplest form.

We convert the fractions to decimals by dividing $$1$$ by $$2$$ and $$1$$ by $$5$$. Thus, $$\dfrac{1}{2}=0.5$$ and $$\dfrac{1}{5}=0.2$$

Now, we compare both the ratios.

$$0.5>0.2$$

$$\Rightarrow\dfrac{2}{4}>\dfrac{1}{5}$$

$$\Rightarrow\text{2 to 4 > 1 to 5}$$

Hence, option (A) is correct.

### Which one of the following ratios is greater,  $$\text{2 to 4 or 1 to 5}$$ ?

A

$$\text{2 to 4}$$

.

B

$$\text{1 to 5}$$

C

$$\text{Both are equal}$$

D

Option A is Correct

# Statement Form of Ratio

• A ratio can be written in the form of statements.
• For example: A bag contains white and black balls. The ratio of white balls to the total balls is $$5:15$$.

Total balls $$=15$$, White balls $$=5$$, Black balls $$=15-5=10$$

Here, the ratio can be written in statement form as follows-

(i) For every $$5$$ white balls there are  $$10$$ black balls in the bag.

(ii) By the equivalent ratio, $$\dfrac{5}{10}=\dfrac{10}{20}$$

Thus, there are $$10$$ white balls for every $$20$$ black balls.

(iii) There are $$5$$ white balls for total $$15$$ balls in the bag.

#### In a box, there are red and green squares. The ratio of red squares to green squares is $$7:9$$. Which statement is FALSE?

A There are $$7$$ red squares for every $$9$$ green squares in the box.

B There are $$7$$ red squares for total $$16$$ squares in the box.

C There are $$9$$ green squares for total $$16$$ squares in the box.

D There are $$9$$ red squares for every $$7$$ green squares in the box.

×

Given statement :

The ratio of red squares to green squares is $$7:9$$.

Number of red squares $$=7$$

Number of green squares $$=9$$

Total number of squares in the box $$=7+9=16$$

From the given ratio, we can write :

(i) There are $$7$$ red squares for every $$9$$ green squares in the box.

(ii) There are $$7$$ red squares for total $$16$$ squares in the box.

(iii) There are $$9$$ green squares for total $$16$$ squares in the box.

Hence, option (D) is false.

### In a box, there are red and green squares. The ratio of red squares to green squares is $$7:9$$. Which statement is FALSE?

A

There are $$7$$ red squares for every $$9$$ green squares in the box.

.

B

There are $$7$$ red squares for total $$16$$ squares in the box.

C

There are $$9$$ green squares for total $$16$$ squares in the box.

D

There are $$9$$ red squares for every $$7$$ green squares in the box.

Option D is Correct