CBSE Class 7 Mathematics Comparing Quantities Notes

CBSE Class 7 Comparing Quantities Concepts. Learning the important concepts is very important for every student to get better marks in examinations. The concepts should be clear which will help in faster learning. The attached concepts made as per NCERT and CBSE pattern will help the student to understand the chapter and score better marks in the examinations.

8.1 Comparing Quantities

To compare two quantities, the units must be the same. We are going to discuss the following ways to compare two quantities:

i) Ratios and proportions ii)   Unitary method iii)  Percentage iv)  Profit and Loss v)Simple Interest

8.2  Ratios and Proportion

Ratio: The ratio of two quantities of the same kind and in the same units is a fraction that shows how many times the one quantity is of the other. Thus, the ratio of two quantities a and b (b ≠ 0) is a ÷ b or a/b and is denoted by a : b

In the ratio a : b the quantities a and b are called the terms of the ratio. The former ‘a’ is called the first term or antecedent and the later ‘b is known as the second term or consequent.

Ratio in the simplest form or lowest form: A ratio a : b is said to be in the simplest form if its antecedent a and consequent b are co – prime.

For example, The lowest form of 80 : 32 is 5 : 2.

Note:  i)    In a ratio, we compare two quantities. The comparison becomes meaningless if the quantities being compared  are not of the same king. i.e they are not measured in the  same units. It is just meaningless to compare 20 bags with 200 crows. Therefore, to find the ratio of two quantities, they must be expressed in the same units.

ii) The order of the terms in a ratio a : b is very important. The ratio 3 : 2 is entirely different from  the ratio 2 : 3.

Comparison of ratios: In order to compare two given ratios, follow the following steps.

Step – I  Express each ratio in the simplest fractional form.

Step – II  By finding the L.C.M of denominators convert them in to like fractions

Step – III Now compare the numerator of the like fractions to find which one is smaller and which one is larger.

Example 1:Compare the ratios: 10 : 24 and 3 : 8

Solution:  Step 1: Writing the given ratios in simplest fractional form

Step 2:  Converting into like fractions

L.C.M (12, 8) = 24

Therefore,  and

Step 3: Comparing the numerator 10 > 9

Hence, 10 : 24 > 3 : 8

Equivalent Ratios: A ratio obtained by multiplying or dividing the numerator and denominator of a given ratio by the same number is called a equivalent ratio.

Example 2: Are the ratios 1 : 2 and 2 : 3 equivalent?

Solution:  To check this, we need to know whether

 We have  

We find that , which means that

 Therefore, the ratio 1 : 2 is not equivalent to the ratio 2 : 3

Example 3: Find an equivalent ratio for 2 : 3

Solution: Proportion: An equality of two ratios is called a proportion. Four numbers a, b, c and d are said to be in proportion, if the ratio of the first two is equal to the ratio of the last two, i.e., 

a : b = c : d Symbolically it is denoted by a : b :: c : d . (Read it as a is to b as c is to d or a is to b is in proportion with c is to d).

Here a, b, c and d are the first, second, third and fourth term of the proportion respectively. The first and fourth terms of a proportion are called extreme terms or extremes while the second and third terms are called middle terms or means.

Relation between extremes and means: Product of extremes  = Product of means 
i.e., a × d = b × c

Note: If ad ≠ bc, the a, b, c, d are not in proportion.

Continued Proportion: Three numbers a, b, c are said to be in continued proportion if 
a : b :: b : c i.e., b² = ac

Mean Proportional: If a, b, c are in continued proportional, then b is called the mean proportional between a and b.

Example 4:  Are 36, 49, 6, and 7 in proportion?

Solution:Product of extremes = 36 x 7 = 252

Product of means = 49 x 6 = 294

Product of extremes ≠ Product of means

Hence, they are not in proportion

Example 5: What must be added to the numbers 6, 10, 14 and 22 so that they are in proportion?

Solution: Let the required number be x.

Then, 6 + x, 10 + x, 14 + x and 22 +x are in proportion.

⇒ Product of extremes = Product of means

⇒ (6 + x) (22 +x) = (10 + x)( 14 + x )

⇒ 132 + 22 x + 6 x + x2 = 140 + 14 x +10 x + x2

⇒ 132 + 28x = 140 + 24x

⇒ 28x – 24 x = 140 – 132

⇒ 4x = 8

⇒ x = 2

8.3 Unitary Method

The method of finding first the value of one article from the value of the given number of articles and then the value of the required number of articles is called the unitary method.

Formula and Unitary Method:

                                        Value of a given number of articles
Value of one article =      ————————————————————
                                                 Number of articles

Example 6: 25 workers earn Rs.300 per day. What will be the earnings of 20 workers per day at the same rate?

Solution: Per day earning of 25 workers = Rs. 300

Per day earning of 1 worker = Rs. 300 ÷ 25 = Rs. 12
                                       (using unitary method)

Per day earning of 20 workers = Rs. 12 × 20 = Rs.240

8.4 Percent

The word per cent is an abbreviation of the Latin phrase ‘per centum’ which means per hundred or hundredths.

Percent as fraction: A fraction with its denominator 100 is equal to that per cent, as the numerator.

Example 7: 75/100 = 75 * 1/100 = 75%

Percent as a ratio: A per cent can also be expressed as a ratio with its second term 100 and first term equal to the given per cent
.
Example 8: 13% = 13/100

Conversion of Per cent into fraction:

Step –I: Given x%

Step – II:  x% = x/100 and simplify x/100 to its lowest form

Example 9: Convert 36% as fractions in the simplest form.

Solution: 36% = 36/100 = 9/25

Conversion of fraction into a per cent:

Step – I: Given fraction a/b

Step – II: Multiply the given fraction by 100 and put per cent sign % to obtain the required per cent.

Thus,  a/b = ( a/b * 100 )%

Example 10: Express the following fraction as per cents: 9/20

Solution: 9/20 = (9/20*100)% = (9*5)% = 45%

Note: If required the fractional form can be converted into decimal form and vice versa

Conversion of ratio into percent

Step – I: Given ratio a : b

Step –II: Convert ratio into fractional form a/b

Step –III: Follow the same steps as in conversion of fractions into per cents.

Example 11: Express 6 : 5 as per cents.

Solution: 6:5 = 6/5 = ( 6/5*100 )% = (6*20)% = 120%

Conversion of Per cent into ratio

Step - I: Given x%

Step - II: x% = x/100 and simplify x/100 to its lowest form

Step – III : Express the fraction obtained in Step –II as a ratio.

Example 12: Express 25% as ratios

Solution: 25% = 25/100 = 1/4 = 1:4

Finding percentage of a given number.

Step – I : Given a number, say x and a percentage , say p%

Step – II : Multiply x by p and divide by 100 to obtain the required p% of x.

               i.e. p% of x = P/100*x

Example 13: Find 10% of 350 km.

Solution: 10% of 350 km = 10/100*350km = 35 km

Finding how much per cent one quantity is of another quantity: Let a and b be two numbers and we want to know what per cent of a is b?

Let x% of a be equal to b. Then, x/100 * a = b

Therefore, x = b/a * 100

Thus, b is ( b/a 100 )% of a

Example 14: What per cent of 25 kg is 3.5 kg?

Solution: We know that b is (b/a*100) % of a.

Here a = 25 kg, b = 3.5 kg

(b/a*100)% = (3.5/25 × 100)%

=(3.5*4)%

= 14%

Therezore, 3.5 kg is 14 % 0f 25 kg

Aliter:

Let x% of 25kg be equal to 3.5kg. Then,

x/100*25 = 3.5kg

Therefore, x = 3.5/25*100 = 14

Thus, 3.5kg is 14% of 25 kg.

8.5 Percentage Change

Formulae: i) Increase % = ( Increase / original value * 100 ) %

ii) Decrease % = ( Decrease / original value * 100 ) %

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