1. Fractions, decimals, estimation
Benchmarks (1/3, 1/4, 3/8...), cross-multiply to compare fractions, smart rounding, dominant term, scientific notation.
Core ideas
- Anchor to benchmarks: 1/8 = 0.125, 1/6 = 0.1667, 3/8 = 0.375, 5/6 = 0.8333; compare unknowns to these instead of dividing.
- To compare a/b vs c/d, cross-multiply: compare a*d vs c*b (safe when all values are positive).
- In messy arithmetic, round to friendly numbers and track whether you rounded up or down; the dominant term (largest magnitude) decides the answer when choices are far apart.
- Scientific notation: move one number's exponent to match the other, then compare or add the coefficients.
Worked example 1
Which is greater: 7/9 or 11/14?
Show solution
Cross-multiply. Compare 7*14 = 98 with 11*9 = 99. Since 99 > 98, and the 99 came from 11 (the numerator of 11/14), 11/14 is greater. This avoids long division entirely and works whenever both fractions are positive.
Worked example 2
Estimate the value of (4.02 * 10^8 + 3.9 * 10^6) / (7.98 * 10^5).
Show solution
The dominant term in the numerator is 4.02 * 10^8; the 3.9 * 10^6 term is about 1 percent of it, so write the numerator as roughly 4.06 * 10^8. Divide: 4.06 / 7.98 is about 0.51, and 10^8 / 10^5 = 10^3. So the value is about 0.51 * 10^3, which is about 510. If answer choices are spread out (say 5, 50, 500, 5000), pick 500 without any exact arithmetic.
Practice set
Question 1
The value of (0.507 * 819) / (0.249 * 41.2) is closest to which of the following?
Round each number to a friendly nearby value before doing any arithmetic.
Round 0.507 to 0.5, 819 to 820, 0.249 to 0.25, and 41.2 to 41. The numerator is about 0.5 * 820 = 410 and the denominator is about 0.25 * 41 = 10.25. Then 410 / 10.25 is about 40, so the answer is C.
Question 2
What is the value of (8 * 10^12) / (2 * 10^-3) ?
Handle the coefficients and the powers of 10 separately, and remember what dividing by a negative exponent does.
Divide the coefficients: 8/2 = 4. Divide the powers of 10: 10^12 / 10^-3 = 10^(12 - (-3)) = 10^15. The result is 4 * 10^15, choice B. Choice A comes from subtracting 3 instead of adding it.
Question 3
What is the value of (2^30 - 2^28) / (2^28 - 2^26) ?
Factor the smallest power of 2 out of the top and out of the bottom before dividing.
Factor the numerator: 2^30 - 2^28 = 2^28 (2^2 - 1) = 2^28 * 3. Factor the denominator: 2^28 - 2^26 = 2^26 (2^2 - 1) = 2^26 * 3. The quotient is (2^28 * 3) / (2^26 * 3) = 2^2 = 4, choice C. Subtracting exponents directly, as in 2^(30-28), gives the trap answer 2 only if the subtraction inside each parenthesis is ignored.
Question 4
Which of the following lists the fractions 7/9, 11/14, and 4/5 in increasing order?
Cross-multiply each pair rather than hunting for a common denominator.
Compare 7/9 and 11/14 by cross-multiplying: 7 * 14 = 98 and 9 * 11 = 99, so 7/9 < 11/14. Compare 11/14 and 4/5: 11 * 5 = 55 and 14 * 4 = 56, so 11/14 < 4/5. The increasing order is 7/9, 11/14, 4/5, choice A.