(c) The sum of the ages of Vivek and his younger brother Amit is 47 years. The product of their ages in
years is 550. Find their ages.
[3]
(a) Find the 25th
-5
Answers
Given :
- Sum of ages of Vivek and his younger brother Amit = 47 years
- Product of their ages = 550 years
To find :
- Age of Vivek
- Age of Amit
Concept :
To find the ages of Vivek and Amit, firstly assume their ages as any variable. Let it be x and y. Then according to the conditions given in the question we will solve equations and find the value of the variables.
Solution :
Let,
- The present age of Vivek = x years
- The present age of Amit = y years
According to the first condition, mentioned in the question :
→ Age of Vivek + Age of Amit = 47
→ x + y = 47 ………………(1)
According to the second condition, mentioned in the question :
→ Age of Vivek × Age of Amit = 550
→ x × y = 550
→ xy = 550 ………………(2)
Taking equation (1)
→ x + y = 47
Transposing y to the right hand side.
→ x = 47 - y ………………(3)
Substitute the value of x in equation (2) :
→ xy = 550
→ (47 - y)y = 550
→ 47y - y² = 550
Transposing 550 to the left hand side.
→ 47y - y² - 550 = 0
→ - y² + 47y - 550 = 0
Quadratic equation is formed now.
Product = -y² × - 550 = + 550y²
★ Product = 550 y² = 0
→ - y² + 47y - 550 = 0
Take (-) common.
→ - (y² - 47y + 550) = 0
→ - (y² - 22y - 25y + 550) = 0
→ - [y(y - 22) - 25(y - 22)] = 0
→ - (y - 25)(y - 22) = 0
CASE I
→ - (y - 25) = 0
→ - y + 25 = 0
→ - y = - 25
→ y = 25
CASE II
→ - (y - 22) = 0
→ - y + 22 = 0
→ - y = - 22
→ y = 22
When y = 25
Substitute the value of y in equation (1)
→ x + y = 47
→ x + 25 = 47
→ x = 47 - 25
→ x = 22
When y = 25, the value of x = 22 :
Therefore,
- Present age of Vivek = x years = 22 years
- Present age of Amit = y years = 25 years
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When y = 22
Substitute the value of y in equation (2)
→ x + y = 47
→ x + 22 = 47
→ x = 47 - 22
→ x = 25
When y = 22, the value of x = 25 :
Therefore,
- Present age of Vivek = x years = 25 years
- Present age of Amit = y years = 22 years
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VERIFICATION :
To verify the ages of Vivek and Amit, add their ages if they sum up to 47 years (as mentioned in the question) then the values are right.
→ Age of Vivek + Age of Amit
→ 22 + 25
→ 47
Age of Vivek + Age of Amit = 47 years