Answer: First number = 50, Second number = 30
Solution:
Assuming that the numbers are integers, let p denote the first number and q the second number.
Given, sum of the two numbers is 80.
∴ p+q = 80………………………………………………………………………………………………(1)
⇒ q = 80 - p ………………………………………………………….………………………………..(2)
Also given, 3 times the first number (p) is same as 5 times the second number (q).
∴ 3p = 5q ………………………………………………………………………………………………..(3)
Simultaneous solution of equations (1) and (3) in the two variables p and q will furnish the answer.
Method 1: Substitution Method
Substituting for q from (2) in (3),
3p = 5(80-p)
Or, 3p = 5x80 - 5p = 400 - 5p
Or, 8p = 400 (Transposing -5p from right to left and simplifying)
Or, 8p/8 = 400/8 (Dividing both sides by 8)
Or, 1.p = 50x8/8 (Factoring 400)
⇒ p = 50
Substituting for p = 50 in (2),
q = 80 - 50
⇒ q = 30
Method 2: Elimination Method
Rewriting, (2)→
3p - 5q = 0 ……………………………………………………………………………………………..…(4)
Now 3 x equation(1) - equation(4) →
3p + 3q -3p + 5q = 3x80 - 0
Or, 8q = 240
Dividing both sides by 8 and simplifying give
q = 30
Substituting for q=30 in (1),
p+30 = 80
Subtracting 30 from both sides,
p+30–30 = 80–30
Or, p+0 = 50
⇒p = 50
These values are in agreement with those obtained by method 1.
Hence, the first number = 50 and second number = 30 (Answer)
Result Verification:
p = 50, q= 30
∴ p+q = 50+30 = 80 in agreement with equation (1)
3p = 3x50 = 150
5q = 5x30 = 150
Step-by-step explanation:
