A reliable method
Name the original whole, apply the first fraction, then use the new remainder as the second base. Keep exact fractions until the final calculation.
The common trap
Subtracting two fractions of different bases treats unlike quantities as if they were comparable. Complex fractions require dividing by the entire denominator expression.
Check the conclusion
Check a fraction equation by substitution and verify that all denominators are nonzero. A percentage estimate can help assess size without replacing the exact calculation.
Contrast the cases
If travel uses one quarter of a $120 budget, $90 remains. Supplies using one third of that remainder cost $30, leaving $60. Subtracting 1/4 and 1/3 of the original budget would leave $50 and violates the word “remaining.” In a complex fraction such as (1/3)/(1/3+1/4), the whole sum 7/12 is the divisor. The result is 4/7, not the denominator itself.
Transfer and explain
Transfer: spend 1/5 of $150, then 1/4 of the remainder. The final amount is $90. Recover both expenses, $30 and $30, and verify that expenses plus remainder equal the original budget.
What is the value of (1/3) ÷ (1/3 + 1/4)?
- The denominator is (4 + 3)/12.
- Divide by this fraction by multiplying by its reciprocal.
- Cancellation gives 4/7.
Compare the other answer choices
Returns the denominator expression itself.
Returns only the outside numerator.
Reverses which original reciprocal contributes to the numerator.
Multiplies the inner fractions instead of adding and dividing.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GMAT exam contentSiyavula Grade 10 mathematics foundation reference