Here we will learn how to find the problems on ratios in simplest form. In order to express a ratio in the simplest form, we find the HCF of the terms and divide each term by the HCF.

We know, a ratio must always be expressed in its lowest terms or simplest form. A ratio is said to be in the simplest form if the first term or first quantity (antecedent) and the second term or second quantity (consequent) have no common factor other than 1.

Find the ratio of each of the following in simplest form:

**(i) 30 and 15**

= 30 : 15

First we need to convert the given ratio into fraction,

= 30/15, [divide both the numerator and denominator by 15 since, the h.c.f. of 30 and 15 is 15]

= 2/1

= 2 : 1

**(ii) 60 and 48**

= 60 : 48

First we need to convert the given ratio into fraction,

= 60/48 (divide both the numerator and denominator by 12 since, the h.c.f. of 60 and 48 is 12)

= 5/4

= 5 : 4

**(iii) 8 kg and 10 kg**

= 8 kg : 10 kg

= (8 kg)/(10 kg), [divide both the numerator and denominator by 2 since, the h.c.f. of 8 and 10 is 2]

= 4/5

= 4 : 5

Now, we will solve different types of problems on ratios in simplest form where both the quantities in different units. So, before finding the required ratio, we shall have to express both the quantities in the same units.

**(iv) 3 kg to 2000 gm**

= 3 kg : 2000 gm

= (3 kg)/(2000 gm)

We know, 1 kg = 1000 gm, 3 kg = 3 × 1000 gm = 3000 gm,

= (3000 gm)/(2000 gm), [divide both the numerator and denominator by 1000 since, the h.c.f. of 3000 and 2000 is 1000]

= 3/2

= 3 : 2

**(v) 750 gm to 2 kg 250 gm**

= 750 gm : 2 kg 250 gm

= (750 g)/(2 kg 250 gm)

We know, 1 kg = 1000 gm, 2 kg = 2 × 1000 gm = 2000 gm,

= (750 gm)/(2000 gm + 250 gm)

= 750/2250, [divide both the numerator and denominator by 750 since, the h.c.f. of 750 and 2250 is 750]

= 1/3

= 1 : 3

**(vi) 3 hours to 75 minutes**

= 3 hours : 75 minutes

= (3 hours)/(75 minutes)

We know, 1 hour = 60 minute, 3 hours = 3 × 60 minutes = 180 minutes,

= (180 minutes)/(75 minutes)

= 180/75

= 12/5

= 12 : 5

**(vii) 2 hours 15 minutes to 45 minutes**

= 2 hours 15 minutes : 45 minutes

= (2 hours 15 minutes)/(45 minutes)

We know, 1 hour = 60 minute, 2 hours = 2 × 60 minutes = 120 minutes,

= (120 + 15 minutes)/(45 minutes)

= 135/45

= 3/1

= 3 : 1

**(viii) 10 months and 2 years**

= 10 months : 2 years

= (10 months)/(2 years)

We know, 1 year = 12 months, 2 years = 12 × 2 months = 24 months,

= (10 months)/(24 months)

= 10/24

= 5/12

= 5 : 12

Thus, from the above problems on ratios in simplest form we can understand that the two quantities can be compared when they are of the same kind. We can compare the ages of two persons, but we cannot compare the age of one person with, say, health or wealth of another person. Similarly, length and width can be compared becomes both the quantities are measures of length. The measurements must also be in same unit for comparison.

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