What number must be subtracted from both the numerator and denominator of the fraction 27/35 so that it becomes 2/3?
11
Finding the Number to Change a Fraction
The problem asks us to find a specific number. When this number is subtracted from both the top part (numerator) and the bottom part (denominator) of the fraction 27/35, the fraction changes into 2/3. Let's call the number we are looking for '\(x\)'.
Setting Up the Equation
Based on the problem description, we can write this relationship as an equation:
The original fraction is \(\frac{27}{35}\).
When we subtract \(x\) from the numerator, it becomes \(27 - x\).
When we subtract \(x\) from the denominator, it becomes \(35 - x\).
The new fraction is \(\frac{27 - x}{35 - x}\).
We are told this new fraction is equal to \(\frac{2}{3}\).
So, the equation is:
\[ \frac{27 - x}{35 - x} = \frac{2}{3} \]
Solving the Equation for \(x\)
To solve this equation, we can use cross-multiplication. This means we multiply the numerator of one fraction by the denominator of the other fraction and set them equal.
Multiply the numerator of the left side (\(27 - x\)) by the denominator of the right side (3):
\[ 3 \times (27 - x) \]
Multiply the numerator of the right side (2) by the denominator of the left side (\(35 - x\)):
\[ 2 \times (35 - x) \]
Set these products equal to each other:
\[ 3(27 - x) = 2(35 - x) \]
Now, we distribute the numbers outside the parentheses:
\[ (3 \times 27) - (3 \times x) = (2 \times 35) - (2 \times x) \]
\[ 81 - 3x = 70 - 2x \]
The goal is to get \(x\) by itself on one side of the equation. We can do this by moving terms around.
Add \(3x\) to both sides of the equation:
\[ 81 - 3x + 3x = 70 - 2x + 3x \]
\[ 81 = 70 + x \]
Subtract 70 from both sides of the equation:
\[ 81 - 70 = 70 + x - 70 \]
\[ 11 = x \]
So, the number that must be subtracted from both the numerator and the denominator is 11.
Verifying the Solution
Let's check if subtracting 11 from both 27 and 35 results in the fraction 2/3.
New numerator: \(27 - 11 = 16\)
New denominator: \(35 - 11 = 24\)
The new fraction is \(\frac{16}{24}\).
Now, we simplify the fraction 16/24. We can find the greatest common divisor (GCD) of 16 and 24, which is 8. Divide both the numerator and the denominator by 8:
\[ \frac{16 \div 8}{24 \div 8} = \frac{2}{3} \]
The simplified fraction is indeed 2/3. This confirms that our value for \(x\) is correct.
Revision Table: Key Concepts
| Concept | Explanation |
|---|---|
| Numerator | The top number in a fraction, representing the number of parts taken. |
| Denominator | The bottom number in a fraction, representing the total number of equal parts in the whole. |
| Equation | A mathematical statement that shows two expressions are equal, connected by an equals sign (=). |
| Cross-multiplication | A method used to solve equations involving fractions. It involves multiplying the numerator of one fraction by the denominator of the other. |
Additional Information: Solving Fraction Problems
Fraction problems often involve setting up an equation based on the given information. Here are some common scenarios and tips:
- Adding/Subtracting a number: If a number is added to or subtracted from the numerator or denominator, represent the new fraction algebraically, like we did with \(\frac{27 - x}{35 - x}\).
- Simplifying Fractions: Always simplify fractions to their lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD).
- Solving Algebraic Equations: Once you have an equation (often by cross-multiplying), use inverse operations (addition/subtraction, multiplication/division) to isolate the variable (like \(x\)). Remember to perform the same operation on both sides of the equation to keep it balanced.
- Verification: It's a good practice to substitute your answer back into the original problem or equation to ensure it makes sense and satisfies the conditions.
Understanding these basic steps will help you tackle many types of fraction-related algebraic problems.
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