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Lesson Example Discussion Quiz: Class Homework

Example

Title: Ratios

Grade: 8-a Lesson: S2-L4

Explanation: Here are some examples of the topic with images and steps in sequence.

Examples:

Example: 1a

What should be added to each term of the ratio 5 : 11, so that the ratio becomes 3 : 4?

Given ratio = 5:11
Let the term to be added = k
As per given data = (5 + k):(11 + k) = 3:4
⇒4(5 + k) = 3(11 + k)
⇒ 20 + 4k = 33 + 3k
⇒ k = 13.

Explanation: In the above problem, the given ratio is 5 : 11 and the term to be added should be considered as 'k'. After simplification, 13 is the number that should be added to 5:11 to become 3:4.

Example: 2a

The yearly incomes of two persons are in the ratio 8 : 10 and their yearly expenditures are in the ratio 7 : 9. If each saves $100 per year, find the monthly income of the second person.

Income ratio of the two persons = 8:10
Expenditure ratio = 7:9
Income of first person = 8x
Income of second person = 10x
Expenditure = Income - Savings
Expenditure of 1st person = 8x - 100
Expenditure of 2nd person = 10x - 100
So, we have = \$(8x - 100):(10x - 100)\$ = \$7:9\$
⇒ \$(8x - 100)/(10x - 100)\$ = \$7/9\$
⇒ \$9(8x - 100)=7(10x - 100)\$
⇒ \$72x - 900\$ = \$70x - 700\$
⇒ x = 100
Then,the income of the second person = 10x
⇒ \$10 \times 100 = 1000\$

Explanation: In the above problem, the yearly income of the two persons is in the ratio of 8:10 and monthly expenditure is in the ratio is 7:9. After applying Expenditure = Income - Savings and simplifying the problem, we get the value of yearly income of the second person as $1000.

Example: 3a

Two numbers are in the ratio 7 : 5. If 2 is subtracted from each of them, the ratio becomes 3 : 2. Find the numbers.

Ratio of the two numbers = 7:5
Let x be the number ⇒ 7x : 5x
The ratio if 2 is subtracted from each number = 3 : 2.
\$(7x - 2):(5x - 2)\$ = \$3:2\$
⇒\$(7x - 2)/(5x - 2)\$ = \$3/2\$
⇒\$2(7x - 2)\$ = \$3(5x - 2)\$
⇒ \$14x - 4 = 15x - 6\$
⇒ \$x = 2\$
The numbers are ⇒ 7x = 7 x 2 = 14 ⇒ 5x = 5 x 2 = 10

Explanation: In the above problem, the ratio of two numbers is 7 : 5 and 'x' is considered as the number. The ratio if 2 is subtracted from each number is 3 : 2. After simplification, x is 2 and the numbers are 14 and 10.