Questions 1.
The sum of two numbers is 18 . Find the numbers if their product is 45.
Questions 2.
Find two numbers whose product is 36 and difference is 5.
Answers
Answer:
1. 15 and 3
2. 9 and 4
Step-by-step explanation:
1. Given sum of two numbers is 18 and product is 45
taking prime factorization of 45 we get
45 = 5*3*3
upon which we can get 15*3 gives 45 and the sum is 18. So the numbers are 15 and 3
2. Given two numbers product is 36 and difference is 5
Taking prime factorization we get
36 = 2*2*3*3 = 4*9
We see that 9*4 gives 36 as product and difference gives 5 the numbers are 9 and 4.
ANSWER 1):
Given:
- Sum of 2 numbers = 18
- Product of 2 numbers = 45
To Find:
- The numbers
Solution:
Let the two numbers be x and y.
We are given that,
- x + y = 18
- xy = 45
Transposing x to RHS in (2),
⟹ y = 45/x ------(3)
Substituting the value of y in (1),
⟹ x + y = 18
⟹ x + 45/x = 18
Taking LCM,
⟹ (x² + 45)/x = 18
⟹ x² + 45 = 18x
⟹ x² - 18x + 45 = 0
Splitting the middle term,
⟹ x² - 15x - 3x + 45 = 0
⟹ x(x - 15) - 3(x - 15) = 0
⟹ (x - 3)(x - 15) = 0
⟹ x = 3, 15
Now, substituting the value of x in (3),
⟹ y = 45/x
For x = 3,
⟹ y = 45/x
⟹ y = 45/3
⟹ y = 15
For x = 15,
⟹ y = 45/x
⟹ y = 45/15
⟹ y = 3
Therefore, the numbers are 3 and 15.
ANSWER 2):
Given:
- Difference of 2 numbers = 5
- Product of 2 numbers = 36
To Find:
- The numbers
Solution:
Let the two numbers be x and y.
We are given that,
- x - y = 5
- xy = 36
Transposing x to RHS in (2),
⟹ y = 36/x ------(3)
Substituting the value of y in (1),
⟹ x - y = 5
⟹ x - 36/x = 5
Taking LCM,
⟹ (x² - 36)/x = 5
⟹ x² - 36 = 5x
⟹ x² - 5x - 36 = 0
Splitting the middle term,
⟹ x² - 9x + 4x - 36 = 0
⟹ x(x - 9) + 4(x - 9) = 0
⟹ (x - 9)(x + 4) = 0
⟹ x = 9, -4
Now, substituting the value of x in (3),
⟹ y = 36/x
For x = 9,
⟹ y = 36/x
⟹ y = 36/9
⟹ y = 4
For x = -4
⟹ y = 36/x
⟹ y = 36/-4
⟹ y = -9
Therefore, the numbers are 9 and 4 or -4 and -9.