NCERT Solutions for Class 7 Maths Chapter 8 Exercise 8.1

NCERT Solutions for Class 7 Maths Chapter 8 Exercise 8.1 (Ex. 8.1) Comparing Quantities in Hindi and English Medium updated for 2020-2021. These PDF and Videos Solution are helpful for CBSE Board as well as State board students to clear their doubts.

In class 7 math exercise 8.1, we will study about ratios of two number and application based on daily life situation. All the questions in ex. 8.1 are easy to understand or solve.

Class 7 Maths Chapter 8 Exercise 8.1 Solution

Class: 7Mathematics
Chapter: 8Comparing Quantities
Exercise: 8.1PDF and Video Solution

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Ratio

The ratio of two quantities of the same kind and in the same unit is the fraction that one quantity is of the other.
The ratio a is to b is the fraction a/b, written as a : b.

Terms of a Ratio

In the ratio a : b, we call a the first term or antecedent and b the second term or consequent.
Example:
In the ratio 3 : 4, the first term or antecedent is 3 and the second term or consequent is 4.

An Important Property

A ratio remains unchanged if both of its terms are multiplied by the same non zero quantity.
Let m ≠ 0.
Then, clearly we have:
(i) a/b = ma/mb and therefore, (a : b) = (ma : mb).
(ii) a/b = (a/m)/ (b/m) and therefore, (a : b) = (a/m : b/m).

Ratio in Simplest Form:

The ratio (a : b) is said to be in simplest form if HCF of a and b is 1.

Express the ratio 75 : 125 in simplest form.

HCF of 75 and 125 is 25.
So, 75 : 125 = 75/125
= (75/25)/ (125/25) = 3/5 = 3 : 5
Hence, 75 : 125 in simplest form is 3 : 5

Comparison of ratios:

We compare two ratios in the same manner as we do in case of fractions.We compare two ratios in the same manner as we do in case of fractions.

Which is greater 4 : 5 or 16 : 25?

We have:
4 : 5 = 4/5 and 16 : 25 = 16/25.
LCM of 5 and 25 is 25.
Now, 4/5 = (4/5) x (5/5) = 20/25 and 16 : 25 = 16/25.
Now, 20/25 > 16/25, and hence (4 : 5) > (16 : 25).

Proportion

Four numbers a, b, c, d are said to be in proportion,
if a : b = c : d and we write, a : b : : c : d.



(a). If a : b : : c : d, then

(i) a, b, c, d are respectively known as first, second, third and fourth term.
(ii) a and d are called extremes while b and c are called means.
(iii) (a x d) = (b x c), i. e., product of extremes = product of means.
(iv) d is called the fourth proportional to a, b, c.

(b). If a : b : : b : c, then

(i) a, b, c are said to be in continued proportion.
(ii) c is called third proportion to a, b and fourth proportional to a, b, b.
(iii) a/b = b/c
Or, b2 = ac
Or, b = √ac and, b is called the mean proportional or geometric mean between a and c.

Find the mean proportional between 8 and 18.

Let the mean proportional between 8 and 18 be x.
Then, 8 : x : : x : 18
Or, x X x = 8 X 18.
Or, x2 = 144 = 12 X 12 [So, product of extremes = product of means]
Or, x = 12

Unitary Method

A method in which the value of unit quantity is first obtained to find the value of any required quantity, is called unitary method.

Solving problems on unitary method, we come across two types of variations.

(i) Direct Variation or Direct Proportion
(ii) Inverse Variation or Inverse Proportion

Direct Variation

Two quantities a and b are said to vary directly, if the ratio a/b remains constant.
Examples:
(i) The cost of articles varies directly as the number of articles. (More articles, more cost), (Less articles, less cost)
(ii) The work done varies directly as the number of men at work. (More men at work, more work), (Less men at work, less work)

Direct Variation

Two quantities a and b are said to vary directly, if the ratio a/b remains constant.
Examples:
(i) The cost of articles varies directly as the number of articles. (More articles, more cost), (Less articles, less cost)
(ii) The work done varies directly as the number of men at work. (More men at work, more work), (Less men at work, less work)

If 25 metres of cloth costs Rs. 1575, how many metres of it can be bought for Rs. 2016? For Rs. 1575, cloth bought = 25 m.

For Rs. 1, cloth bought = 25/1575 m
For Rs. 2016, cloth bought = (25/1575) x 2016 m
= 32 m.
Hence, the length of cloth bought for Rs. 2016 is 32 m.

Inverse Variation

Two quantities are said to be inversely proportional to each other, if increase in the value of one quantity results in the decrease of the second quantity and decrease in one quantity results in the increase of other in such a way that their product always remains constant. When two quantities x and y are such that xy remains constant, then we say that x and y vary indirectly.



How do you simplify a ratio?

Ratios can be fully simplified just like fractions. To simplify a ratio, divide all of the numbers in the ratio by the same number until they cannot be divided any more.

How do you calculate ratios and proportions?

Ratios and Proportions:
Proportions: A proportion is simply a statement that two ratios are equal. It can be written in two ways: as two equal fractions a/b = c/d; or using a colon, a:b = c:d.
The following proportion is read as “twenty is to twenty-five as four is to five.”

The length of the shadow of a 3-m-high pole at a certain time of the day is 3.6 m. What is the height of of another pole whose shadow at that time is 54 m long?

If the length of shadow is 3.6 m, height of the pole = 3 m.
If the length of shadow is 1 m, height of the pole = 3/3.6 m
If the length of shadow is 54 m, height of the pole = (3/3.6) x 54 m = 45 m
Hence, the height of the pole is 45 m

NCERT Solutions for Class 7 Maths Chapter 8 Comparing Quantities Exercise 8.1
NCERT Solutions for Class 7 Maths Chapter 8 Comparing Quantities Exercise 8.1 in PDF form free
7 Maths Chapter 8 Comparing Quantities Exercise 8.1 in Hindi
7 Maths Chapter 8 Comparing Quantities Exercise 8.1 in Hindi pdf free