ORIGINAL WORKED EXAMPLE
Apply the idea
Which is greater, 17/18 or 23/24?
- 17/18=1−1/18 and 23/24=1−1/24.
- Because 1/24 is smaller than 1/18, subtracting it leaves more.
- As a cross-check, 17×24=408 and 23×18=414.
Answer 23/24.
GMAT · Arithmetic and number properties · TOPIC LESSON
Fractions, decimals and efficient arithmetic
Original GMAT Problem Solving instruction: reason from constraints, calculate efficiently without a calculator, and check the requested quantity.
BY THE END, YOU CAN
- Explain and apply fraction equivalence and comparison.
- Explain and apply operations and cancellation.
- Explain and apply mixed numbers.
- Explain and apply decimal place value.
- Explain and apply terminating and repeating decimals.
- Explain and apply order of operations.
- Explain and apply estimation and bounds.
- Explain and apply avoiding invalid cancellation.
BEFORE YOU START
- Whole-number arithmetic, factors and positive/negative signs.
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GMAT study through an editorial skill map. It does not reproduce official questions, predict exam content or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
Original GMAT-style worked examples and practice
All passages, examples and exercises here are original. No official GMAC questions, choices or explanations are reproduced. This independent preparation resource is not affiliated with or endorsed by GMAC. GMAT is a trademark of the Graduate Management Admission Council.
01 / UNDERSTAND & APPLY
Equivalent fractions and comparisons
A fraction a/b represents division and requires b≠0. Multiplying numerator and denominator by the same nonzero number preserves the value. To compare fractions, first make denominators positive; then use a common denominator or positive cross products. A negative denominator left untreated can reverse the intended comparison.
Benchmark fractions often beat large products. If two fractions have the same positive numerator and positive denominators, the smaller denominator gives the larger fraction. With the same negative numerator, the smaller positive denominator gives the more negative, hence smaller, fraction: −3/2<−3/4. With numerator zero, both fractions equal zero. With equal positive denominators, compare numerators. A fraction close to 1 can be compared through its deficit from 1.
ORIGINAL PRACTICE · 1
Order 5/8, 2/3 and 7/12 from least to greatest.
Reveal answer & reasoning
7/12 < 5/8 < 2/3.
Use denominator 24: 14/24,15/24,16/24. Comparing only numerators would put 2/3 first, but denominators determine piece sizes.
02 / UNDERSTAND & APPLY
Operations and legitimate cancellation
Addition combines equal-sized pieces, so use a common denominator before adding numerators. Multiplication multiplies numerators and denominators. Division by c/d means multiplication by d/c and requires the divisor c/d to be nonzero.
Cancel factors across a product before multiplying to keep numbers small. Cancellation is division of an entire numerator and denominator by the same factor; it never means deleting matching terms from a sum. Keeping calculations as fractions avoids rounding and supports no-calculator work.
ORIGINAL WORKED EXAMPLE
Apply the idea
Evaluate (14/15)×(25/21)÷(5/6).
- Replace division by 5/6 with multiplication by 6/5.
- Cancel 14/21 to 2/3 and 25/15 to 5/3.
- Then (2/3)×(5/3)×(6/5)=12/9=4/3.
Answer 4/3.
ORIGINAL PRACTICE · 1
Evaluate 3/4−(2/3)×(3/8).
Reveal answer & reasoning
1/2.
Multiply first: (2/3)(3/8)=1/4 after cancelling 3 and reducing 2/8. Then 3/4−1/4=1/2. Subtracting before multiplication violates the operation order.
03 / UNDERSTAND & APPLY
Mixed numbers express addition
A positive mixed number 2 3/5 means 2+3/5=13/5, not 2×3/5. Convert to an improper fraction before multiplying or dividing. A negative mixed number −2 3/5 conventionally means −(2+3/5)=−13/5. Write parentheses when signs could be ambiguous.
For addition and subtraction, keeping whole and fractional parts separate can be efficient, provided borrowing is handled exactly. Improper fractions are safer when several operations or negative values appear.
ORIGINAL WORKED EXAMPLE
Apply the idea
A ribbon 7 1/2 metres long is cut into pieces each 3/4 metre long. How many complete pieces result?
- Convert 7 1/2 to 15/2.
- Divide by 3/4: (15/2)×(4/3)=10.
- The division is exact, so no leftover remains.
Answer 10 pieces.
ORIGINAL PRACTICE · 1
Compute 4 1/3−1 5/6.
Reveal answer & reasoning
2 1/2.
Convert to sixths: 26/6−11/6=15/6=5/2. Subtracting fractional parts without borrowing can produce a misleading negative fractional part; improper fractions avoid that bookkeeping.
04 / UNDERSTAND & APPLY
Decimal places are powers of ten
A decimal digit denotes a multiple of a power of ten. Thus 0.047=47/1000. Multiplying by 10 shifts every digit one place toward larger place values; dividing by 10 shifts toward smaller values. Leading zeros do not alter value, whereas zeros between digits do.
For decimal division, multiply dividend and divisor by the same power of ten to make the divisor an integer. Estimate the result first. Dividing a positive number by a number between 0 and 1 increases it; this catches many misplaced decimal points.
ORIGINAL WORKED EXAMPLE
Apply the idea
Evaluate 0.084÷0.006 without a calculator.
- Multiply numerator and denominator of the quotient by 1000.
- This gives 84/6=14.
- Check: 0.006×14=0.084.
Answer 14.
ORIGINAL PRACTICE · 1
What is 0.24×0.015?
Reveal answer & reasoning
0.0036.
24×15=360, and the original factors have five decimal places in total: 360/100000=0.0036. A product of two positive numbers below 1 must be smaller than either factor, ruling out 0.36.
05 / UNDERSTAND & APPLY
Why decimals terminate or repeat
Reduce a fraction first. Its decimal terminates exactly when its positive denominator has no prime factors other than 2 and 5, because a power of ten contains only those primes. A denominator such as 12 may look nonterminating, but 3/12 reduces to 1/4 and terminates.
Other rational fractions have repeating decimal expansions. To convert a repeating decimal, multiply by a power of ten that aligns the repeating blocks, then subtract. The decimal ellipsis must actually specify a repeating pattern; a few displayed digits alone do not prove repetition.
ORIGINAL WORKED EXAMPLE
Apply the idea
Convert 0.272727… to a fraction in lowest terms.
- Let x=0.272727… .
- Then 100x=27.272727… . Subtract x to obtain 99x=27.
- Reduce 27/99 by 9.
Answer 3/11.
ORIGINAL PRACTICE · 1
Which fractions terminate: 7/40, 9/60, 5/18? Give all that do.
Reveal answer & reasoning
7/40 and 9/60.
40=2³×5, so 7/40 terminates. 9/60 reduces to 3/20, whose denominator uses only 2 and 5. The reduced denominator of 5/18 contains 3, so it repeats. Testing before reduction gives the wrong verdict for 9/60.
06 / UNDERSTAND & APPLY
Order of operations and grouping
Evaluate parentheses, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication does not automatically precede division. A fraction bar groups the whole numerator and whole denominator.
When copying a verbal or printed expression, preserve grouping. The expression 12/(2+4) equals 2, whereas 12/2+4 equals 10. A reliable check is to state which quantity each operation acts on before performing it.
ORIGINAL WORKED EXAMPLE
Apply the idea
Evaluate 18÷3×2−(5−3)².
- Inside parentheses, 5−3=2, and 2²=4.
- From left to right, 18÷3×2=6×2=12.
- Finally 12−4=8.
Answer 8.
ORIGINAL PRACTICE · 1
Evaluate (3+5)/(2×4)+2³.
Reveal answer & reasoning
9.
The fraction is 8/8=1, and 2³=8; the sum is 9. Treating only 5 as part of the numerator ignores the grouping supplied by the fraction bar.
07 / UNDERSTAND & APPLY
Estimation is a controlled inequality
Use estimates to eliminate choices only when the uncertainty is smaller than the gap between remaining options. For positive products, replacing each factor by a lower bound gives a lower bound; replacing by upper bounds gives an upper bound. For a positive quotient, increase the denominator to obtain a lower bound.
Rounding every input upward does not always make a quotient larger, since an increased denominator works in the opposite direction. An estimate is not an exact answer unless bounds isolate the required integer or answer choice. Keep exact fractions when options are close.
ORIGINAL WORKED EXAMPLE
Apply the idea
Without long division, show that 1,998/49 lies between 40 and 41.
- 49×40=1,960, which is less than 1,998.
- 49×41=2,009, which is greater than 1,998.
- Divide the strict inequalities by positive 49.
Answer 40 < 1,998/49 < 41.
ORIGINAL PRACTICE · 1
A positive quantity is 19.8×5.1. Which is closest: 10, 50, 100, 150 or 200?
Reveal answer & reasoning
100.
20×5=100 is a useful estimate; the exact product is (20−0.2)×5.1=102−1.02=100.98. The answer choices are far enough apart that a small rounding error cannot change the selection.
08 / UNDERSTAND & APPLY
Cancel factors, never isolated addends
In (ab)/(ac), the common factor a can be cancelled only if a≠0. In (a+b)/(a+c), the symbol a is part of each sum, not a factor of each whole expression, so deleting it changes the value. To cancel, factor first and retain the original domain exclusions.
Numerical substitution is an efficient way to refute an alleged identity: one allowed counterexample is enough. It cannot prove an identity for all inputs. Algebraic factoring provides the general proof.
ORIGINAL WORKED EXAMPLE
Apply the idea
Simplify (x²−9)/(x−3), stating every restriction.
- The original denominator excludes x=3.
- Factor the numerator as (x−3)(x+3).
- For x≠3, divide numerator and denominator by x−3.
Answer x+3, for x≠3.
ORIGINAL PRACTICE · 1
Is (x+6)/(x+2)=3 for every allowed x? If not, when is it true?
Reveal answer & reasoning
No; it is true only for x=0.
The denominator requires x≠−2. Multiplying the equation by x+2 gives x+6=3x+6, hence x=0, which is allowed. Cancelling the x addends would falsely claim the equality for all x.
KEEP THE BIG IDEAS
Quick revision
- Make denominators positive before comparing by cross products.
- Cancel common factors, not addends.
- Reduce a fraction before predicting its decimal form.
- Use bounds to justify elimination; use exact arithmetic when choices are close.
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