By what percent will a fraction increase if its numerator is decreased by 10% and its denominator is decreased by 50%?
original fraction: x/y
new fraction:
.9x / .5y = 1.8 x/y
so, it increased by 80%
To calculate the percentage increase of a fraction when its numerator and denominator are decreased, we need to follow these steps:
Step 1: Start with the original fraction.
Step 2: Calculate the new numerator by decreasing the original numerator by 10%.
Step 3: Calculate the new denominator by decreasing the original denominator by 50%.
Step 4: Calculate the new fraction using the new numerator and denominator.
Step 5: Calculate the difference between the new fraction and the original fraction.
Step 6: Calculate the percentage increase by dividing the difference by the original fraction and multiplying by 100%.
Let's apply these steps to the given question.
Original fraction = numerator / denominator
Step 1: Start with the original fraction.
Let's assume the original fraction is a/b.
Step 2: Calculate the new numerator by decreasing the original numerator by 10%.
New numerator = original numerator - (original numerator * 10%)
New numerator = a - (a * 10/100)
New numerator = a - (a / 10)
New numerator = (10a - a) / 10
New numerator = 9a / 10
Step 3: Calculate the new denominator by decreasing the original denominator by 50%.
New denominator = original denominator - (original denominator * 50%)
New denominator = b - (b * 50/100)
New denominator = b - (b / 2)
New denominator = (2b - b) / 2
New denominator = b / 2
Step 4: Calculate the new fraction using the new numerator and denominator.
New fraction = (9a / 10) / (b / 2)
New fraction = (9a / 10) * (2 / b)
New fraction = (9a * 2) / (10 * b)
New fraction = 18a / (10b)
New fraction = (9a / 5b)
Step 5: Calculate the difference between the new fraction and the original fraction.
Difference = New fraction - Original fraction
Difference = (9a / 5b) - (a / b)
Difference = ((9a - 5a) / (5b)) / (b)
Difference = (4a / 5b) / (b)
Difference = 4a / (5b^2)
Step 6: Calculate the percentage increase by dividing the difference by the original fraction and multiplying by 100%.
Percentage increase = (Difference / Original fraction) * 100%
Percentage increase = ((4a / (5b^2)) / (a / b)) * 100%
Percentage increase = ((4a / (5b^2)) * (b / a)) * 100%
Percentage increase = (4b / (5b^2)) * 100%
Percentage increase = (4 / 5b) * 100%
Percentage increase = (4 / 5b) * 100
Therefore, the fraction will increase by (4 / 5b) * 100%.