• 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.

• 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 \(\dfrac{11}{3}\).

• 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, \(3:6=0.5\)

• 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 \(2:4<4:5\).

• 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.