Informative line

Comparison Of Ratios

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.

Illustration Questions

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 \(\dfrac{11}{3}\).

Illustration Questions

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, \(3:6=0.5\)

Illustration Questions

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 \(5\).

image

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 \(2:4<4:5\).

Illustration Questions

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 \(5\).

image

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.

Illustration Questions

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

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