How to Find the Number to Subtract from 3/4 to Get 2/3

Mathematics often presents us with puzzles and challenges that require simple yet precise solutions. One such problem involves determining the number to subtract from 3/4 to obtain 2/3. This task appears straightforward but requires a clear understanding of basic arithmetic and algebra.

Simple Subtraction and Verification

First, let's find the difference between 3/4 and 2/3. The subtraction involves aligning the denominators:

Step 1: Find the common denominator, which is 12.

Step 2: Convert 3/4 9/12 and 2/3 8/12.

Step 3: Subtract 8/12 from 9/12.

Therefore, 9/12 - 8/12 1/12.

Hence, to get from 3/4 to 2/3, you must subtract 1/12.

Algebraic Method

To find the unknown number X to be subtracted, we set up the equation:

Step 1: Represent the unknown number as X.

Step 2: Set up the equation: 3/4 - X 2/3.

Step 3: Solve for X:

Convert both fractions to a common denominator, which is 12. Thus, 3/4 9/12 and 2/3 8/12.

Step 4: The equation becomes 9/12 - X 8/12.

Step 5: Solve for X: X 9/12 - 8/12 1/12.

General Form of the Problem

Let the common factor to be subtracted from both the numerator and denominator be n. Given the ratio 3:4 needs to be converted to 2:3, we can write:

Step 1: Let the original fraction be 3m/4m.

Step 2: Subtract n from both numerator and denominator: (3m - n)/(4m - n) 2/3.

Step 3: Cross-multiply to solve for n:

Step 4: 9m - 2(3m - 2n) 8m - 3m.

Step 5: Simplify: 9m - 6m 6n 8m - 3m.

Step 6: Combine like terms: 3m 6n 5m.

Step 7: Simplify further: 6n 2m.

Step 8: Solve for n: n m/3.

Verification

To verify the solution, substitute n 1/3 into the original fraction:

Step 1: Let m 1. So, the original fraction is 3/4.

Step 2: Subtract 1/3 from both numerator and denominator: (3 - 1/3)/(4 - 1/3).

Step 3: Simplify: (9 - 1)/(12 - 1) 8/11.

Step 4: Verify the result: 8/11 ≠ 2/3.

Hence, the correct number to subtract is 1/12, as previously calculated.

Conclusion

::Subtracting the correct fraction is crucial to maintain mathematical consistency. Whether using simple arithmetic, algebra, or a general form, the solution to find the number to subtract from 3/4 to get 2/3 is consistently 1/12.

Additional Resources

1. Step-by-Step Guide to Solving Fraction Equations * [URL link to guide]

2. Video Tutorials on Ratio and Proportion * [URL link to video tutorials]

3. Practice Problems on Fraction Subtraction * [URL link to practice problems]