A fraction was multiplied by [tex]$\frac{5}{6}$[/tex] to get [tex]$\frac{25}{48}$[/tex]. What was the original fraction?
Answer (2)
The original fraction that was multiplied by 6 5 to get 48 25 is 8 5 .
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Set up the equation: x ⋅ 6 5 = 48 25 .
Multiply both sides by the reciprocal of 6 5 , which is 5 6 : x = 48 25 ⋅ 5 6 .
Simplify the fraction: x = 240 150 .
Reduce the fraction to its simplest form: x = 8 5 .
The original fraction is 8 5 .
Explanation
Understanding the Problem Let's analyze the problem. We are given that a fraction, which we'll call x , was multiplied by 6 5 to obtain 48 25 . Our goal is to find the original fraction x .
Setting up the Equation We can set up the equation:
x × 6 5 = 48 25
To solve for x , we need to isolate it by multiplying both sides of the equation by the reciprocal of 6 5 , which is 5 6 .
Solving for x Now, let's multiply both sides by 5 6 :
x = 48 25 × 5 6
To simplify this, we can multiply the numerators and the denominators:
x = 48 × 5 25 × 6 = 240 150
Now, we simplify the fraction by finding the greatest common divisor (GCD) of 150 and 240.
Simplifying the Fraction We can simplify the fraction 240 150 by dividing both the numerator and the denominator by their greatest common divisor, which is 30:
x = 240 ÷ 30 150 ÷ 30 = 8 5
So, the original fraction was 8 5 .
Final Answer Therefore, the original fraction is 8 5 .
Examples
Imagine you're baking a cake and the recipe calls for a certain amount of flour. However, you only have a measuring cup that's a fraction of the size needed. If you know that using 6 5 of your measuring cup gives you 48 25 of the total flour required, you can use this math to figure out what fraction of the total flour your measuring cup holds. This type of problem helps in scaling recipes or adjusting quantities in various real-life situations where proportional adjustments are needed.
