Informative line

Problems Involving Ratios

Checking whether the Ratios are Equivalent or Not, Using the Cross Product Method

Equivalent ratios

Those ratios whose simplified forms are same 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.

The cross multiplication method is used to compare two ratios by multiplying the numerator of one ratio to the denominator of another ratio.

Here, we are comparing two ratios to find out whether they are equivalent or not.

For example: \(3 : 2\) and \(12 : 8\)

Follow the steps given below:

Step 1: Write the ratios in fraction form.

\(3 : 2=\dfrac {3}{2},\;12:8=\dfrac {12}{8}\)

Step 2: Write each fraction on each side of the \(\stackrel{?}{=}\) symbol.

\(\dfrac {3}{2} \stackrel{?}{=} \dfrac {12}{8}\)

 

 

 

 

Step 3: Apply cross multiplication method.

\(3×8\stackrel{?}{=}12×8\)

\(24\stackrel{?}{=}24\)

\(24=24\)

Hence, \(\dfrac {3}{2}\) and \(\dfrac {12}{8}\) are equivalent ratios.

 

 

Illustration Questions

Which one of the following pairs represents the equivalent ratio?

A \(3:4\) and \(2:4\)

B \(2:5\) and \(5:4\)

C \(7:13\) and \(2:12\)

D \(2:5\) and \(8:20\)

×

For option (A),

Writing the ratios in the fraction form.

\(3:4=\dfrac {3}{4}\),  \(2:4=\dfrac {2}{4}\)

Writing each fraction to each side of the \(\stackrel{?}{=}\) symbol.

\(\dfrac {3}{4}\stackrel{?}{=}\dfrac {2}{4}\)

Applying the cross-multiplication method:

image

\({3}\times{4}\stackrel {?}{=}{2}\times{4}\)

\(12\stackrel {?}{=}8\)

\(12\neq8\)

\(\therefore\;\;3:4\) and \(2:4\) are not equivalent ratios.

Hence, option (A) is incorrect.

For option (B),

Writing the ratios in the fraction form

\(2:5=\dfrac {2}{5}\)\(5:4=\dfrac {5}{4}\)

Writing each fraction to each side of the \(\stackrel{?}{=}\) symbol.

\(\dfrac {2}{5}\stackrel{?}{=}\dfrac {5}{4}\)

Applying the cross-multiplication method:

image

\(2×4\stackrel{?}{=}5×5\)

\(8\stackrel{?}{=}25\)

\(8\neq25\)

\(\therefore\,\;2:5\) and \(5:4\) are not equivalent ratios.

Hence, option (B) is incorrect.

For option (C),

Writing the ratios in the fraction form.

\(7:13=\dfrac {7}{13}\), \(2:12=\dfrac {2}{12}\)

Writing each fraction to each side of the \(\stackrel{?}{=}\) symbol.

\(\dfrac {7}{13}\stackrel{?}{=}\dfrac {2}{12}\)

Applying the cross-multiplication method:

image

\(12×7\stackrel{?}{=}13×2\)

\(84\stackrel{?}{=}26\)

\(84\neq26\)

\(\therefore\)  \(7:13\) and \(2:12\) are not equivalent ratios.

Hence, option (C) is incorrect.

For option (D),

Writing the ratios in the fraction form

\(2:5=\dfrac {2}{5}\)\(8:20=\dfrac {8}{20}\)

Writing each fraction to each side of the \(\stackrel{?}{=}\) symbol.

\(\dfrac {2}{5}\stackrel{?}{=}\dfrac {8}{20}\)

Applying the cross-multiplication method:

image

\(2×20\stackrel{?}{=}5×8\)

\(40\stackrel{?}{=}40\)

\(40=40\)

\(\therefore\;2:5\) and \(8:20\) are equivalent ratios.

Hence, option (D) is correct.

Which one of the following pairs represents the equivalent ratio?

A

\(3:4\) and \(2:4\)

.

B

\(2:5\) and \(5:4\)

C

\(7:13\) and \(2:12\)

D

\(2:5\) and \(8:20\)

Option D is Correct

Finding the Share of Two Quantities

The ratios are also used in dividing up a whole quantity.

For example:  If we need to divide \($80\) between two friends, Aron and Alex in \(3:2\)

How much amount do Aron and Alex get?

To solve the above problem, we need to follow these steps:

Step 1:  Write the part as a fraction (out of total) for both of them.

The ratio of amount between Aron and Alex \(=3:2\)

Total part \(=3+2=5\)

So, fraction of money for Aron \(=\dfrac {3}{5}\)

Fraction of money for Alex \(=\dfrac {2}{5}\)

Step 2: Calculate the amounts for both of them:

Amount of Money Aron gets \(=\dfrac {3}{5}\) of the total money

\(=\dfrac {3}{5}×80\)

\(=3×16=$48\)

The amount of money Alex gets \(=\dfrac {2}{5}\) of total money

\(=\dfrac {2}{5}×$80\)

\(=2×16=$32\)

  • We can also find the total amount (if asked) by using the above method.

Illustration Questions

Jose and Julie worked together on a project and received a sum of money for the same. They both decided to divide the money in \(2:3\). If Julie received \($150\), how much total money did they receive for the project?

A \($250\)

B \($200\)

C \($300\)

D \($350\)

×

Given:

Ratio of amount between Jose and Julie \(=2:3\)

Amount received by Julie \(=$150\)                   ......(1)

Assume the total sum of money as a variable.

Let the sum of money \(=$x\)

Now, finding the total part i.e.

\(2+3=5\)

Fraction of money for Jose \(=\dfrac {2}{5}\)

Fraction of money for Julie \(=\dfrac {3}{5}\)

Amount of money Julie received

\(=\dfrac {3}{5}\) of \(x\)

\(=\dfrac {3}{5}×x\)       ( \(\because\)  of means multiply )  ......(2)

 

\(\therefore \) from (1) and (2), we get:

\($150=\dfrac {3}{5}×\dfrac {x}{1}\)

\(\dfrac {$150}{1}=\dfrac {3x}{5}\)

Applying the cross multiplication method:

image

\($150×5=3x×1\)

\($750=3x\)

Dividing both the sides by \(3\) ,

\(x=\dfrac {$750}{3}=$250\)

Thus, the total sum of money is \($250\).

Hence, option (A) is correct

Jose and Julie worked together on a project and received a sum of money for the same. They both decided to divide the money in \(2:3\). If Julie received \($150\), how much total money did they receive for the project?

A

\($250\)

.

B

\($200\)

C

\($300\)

D

\($350\)

Option A 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

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

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