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Arithmetic I

Ratio & Partnership

ALL MODULES

CAT 2025 Lesson : Ratio & Partnership - Properties of Ratio

1.1 Applying ratios

Ratios tell us the size of one item relative to the other. For instance, if the ratio of apples to mangoes in a bag is

1 : 4

, then the following can be concluded

1

) For every apple in the bag, there are

4

mangoes. Therefore, for every apple in the bag, there are a total of

5

fruits in the bag.

2

) Number of apples is

\dfrac{1}{4}

25 \%

of the number of mangoes.

3

) Number of mangoes is

4

times or

400 \%

of the number of apples.

4

) Number of apples is

\dfrac{1}{1 + 4} = \dfrac{1}{5}

20 \%

of the total number of fruits.

5

) Number of mangoes is

\dfrac{4}{1 + 4} = \dfrac{4}{5}

80 \%

of the total number of fruits.

In most of the questions pertaining to ratios, the variables we assume for our calculations is what the question requires us to find.

Example 2

In an orchard, if mangoes account for two-fifth of the fruits, and rest of the

240

are apples, then how many mangoes are there?

Solution

Let the total number of fruits be

x

.

Number of mangoes

= \dfrac{2x}{5}

Number of apples

= x - \dfrac{2x}{5} = \dfrac{3x}{5} = 240

⇒

x = 400

Number of mangoes =

400

–

240

160

Answer:

160

Example 3

Rahul who is

1.6

m tall and standing under the sun, casts a shadow that is

1.2

m in length. At that moment, what is the length (in metres) of the shadow cast by a

120

m pole?

Solution

Let the length of the shadow of the pole be

x

.

At any time during the day, the ratio of the height of any standing object to the length of the shadow cast by the object will remain the same.

∴

\dfrac{1.6}{1.2} = \dfrac{120}{x}

⇒

x = 90

Answer:

90

2. Properties of ratios

2.1 Multiplying or dividing by a constant

Multiplication or division by a constant leaves the ratio unchanged.

a : b = ka : kb = \dfrac{a}{k} : \dfrac{b}{k}

2.2 Ratio of 2 terms as a Fraction, percentage and decimal

Only ratios with 2 terms can be expressed as fractions. These fraction can in turn be expressed as a decimal or percentage.

5 : 8 = \dfrac{5}{8}

m : n = \dfrac{m}{n}

For instance, if there are

30

pens and

50

pencils, the ratio of pens to pencils can be expressed in any of the following ways.

3 : 5 = \dfrac{3}{5} = 0.6 = 60 \%

2.3 Scaling up and down of Ratios

If we multiply the numerator and denominator by a number greater than

1

, then we are scaling up the ratio.

For example,

\dfrac{3}{5} = \dfrac{3 \times 4}{5 \times 4} = \dfrac{12}{20}

. Here we have scaled up

\dfrac{3}{4}

\dfrac{12}{20}

.

If we divide the numerator and denominator by a number greater than

1

, then we are scaling down the ratio.

For example,

\dfrac{8}{10} = \dfrac{8 \div 2}{10 \div 2} = \dfrac{4}{5}

. Here we have scaled down

\dfrac{8}{10}

\dfrac{4}{5}

2.4 Simplifying Ratios with Fractions

To simplify a ratio with more than two fractions, we multiply the ratio with the LCM of the denominators. This applies for ratios with any number of terms.

For e.g., to find the ratio of

\dfrac{3}{2}, \dfrac{5}{4}

and

\dfrac{7}{6}

, we first find the LCM

(2, 4, 6) = 12

\dfrac{3}{2} : \dfrac{5}{4} : \dfrac{7}{6}

= \dfrac{3}{2} \times 12 : \dfrac{5}{4} \times 12 : \dfrac{7}{6} \times 12

= 18 : 15 : 14

For ratios with exactly 2 terms, we can simplify by cross-multiplying the numerators with the denominators.

\dfrac{a}{b} : \dfrac{c}{d} = ad : bc

For example,

\dfrac{3}{5} : \dfrac{8}{11} = 3 \times 11 : 8 \times 5 = 33 : 40

Example 4

Hazel had

3

kg of wheat and

4

kg of rice. She gave

\dfrac{3}{7}^{\text{th}}

of the wheat and

\dfrac{4}{9}^{\text{th}}

of the rice to Henza. What is the ratio of wheat to rice that Henza received?

(1)

21 : 36

(2)

3 : 4

(3)

27 : 28

(4)

81 : 112

Solution

Ratio of wheat to rice that Henza received

= \dfrac{3}{7} \times 3 : \dfrac{4}{9} \times 4

\dfrac{9}{7} : \dfrac{16}{9}

= \dfrac{9}{7} \times 63 : \dfrac{16}{9} \times 63 = 81 : 112

Alternatively (Recommended for ratios with 2 terms)

\dfrac{9}{7} : \dfrac{16}{9} = 9 \times 9 : 16 \times 7 = 81 : 112

Answer: (4)

81 : 112

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