A man is 5 years older than his wife the product of their ages is 750 find their ages
Let's represent the man's age as "x" and the wife's age as "y". We are given that the man is 5 years older than his wife, so we can write an equation:
x = y + 5
We are also given that the product of their ages is 750:
x * y = 750
Now we can substitute the value of x from the first equation into the second equation:
(y + 5) * y = 750
Expanding the equation:
y^2 + 5y = 750
Rearranging the equation to make it a quadratic equation:
y^2 + 5y - 750 = 0
Now we can factor this quadratic equation:
(y + 30)(y - 25) = 0
Setting each factor equal to zero:
y + 30 = 0 or y - 25 = 0
Solving for y:
y = -30 or y = 25
Since age cannot be negative, we can discard the solution y = -30. Therefore, the wife's age is 25.
Now we can substitute this value into the first equation to find the man's age:
x = y + 5
x = 25 + 5
x = 30
So, the husband is 30 years old and the wife is 25 years old.
Let's solve this step-by-step:
Step 1: Let's assume the wife's age is "x" years. Since the man is 5 years older than his wife, the man's age would be "x + 5" years.
Step 2: The product of their ages is 750, so we can set up the equation: x * (x + 5) = 750.
Step 3: Expanding the equation, we get x^2 + 5x = 750.
Step 4: Rearranging the equation in standard form, we have x^2 + 5x - 750 = 0.
Step 5: We can now solve this quadratic equation. Factoring it would be quite complicated, so let's use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = 1, b = 5, and c = -750. Substituting these values into the quadratic formula:
x = (-5 ± √(5^2 - 4(1)(-750))) / (2*1)
Step 6: Simplifying further, we have x = (-5 ± √(25 + 3000)) / 2.
Step 7: Continuing, we get x = (-5 ± √3025) / 2.
Step 8: The square root of 3025 is 55, so x = (-5 ± 55) / 2.
Step 9: We have two possible solutions: (1) x = (-5 + 55) / 2 = 50 / 2 = 25, or (2) x = (-5 - 55) / 2 = -60 / 2 = -30.
Step 10: Since we are talking about ages, we disregard the negative value, so the wife's age is 25 years.
Step 11: The man's age is 25 + 5 = 30 years.
Therefore, the wife's age is 25 years and the man's age is 30 years.