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Ratios and proportions; distance-rate-time — ISEE Upper practice questions

Ratios and proportions; distance-rate-time on the ISEE Upper Mathematics Achievement section covers equivalent ratios, similar figures, unit rates, and the relationship among distance, rate, and time. A student must set up a proportion, scale corresponding sides, convert a linear scale into an area scale, and find an unknown price or production total from a given rate. Items are typically short word problems with four answer choices. Common traps include applying a side-length scale to area without squaring it, swapping the two quantities in a cost ratio, and treating a same-time machine problem as inverse rather than direct variation. Distance-rate-time questions often hide a unit mismatch or an incorrect average speed.

  • Section: Mathematics Achievement 47 questions on the paper
  • 105 practice questions in the app

Sample questions

Question 1

A triangle has side lengths 6, 8, and 10 and area 24 square units. A second triangle is similar to it, and the side corresponding to the side of length 8 has length 12. What is the area of the second triangle?

  1. A36
  2. B48
  3. C54
  4. D81
Show answer

Answer: C — 54

The side-length scale factor is 128=32\dfrac{12}{8}=\dfrac{3}{2}. Areas scale by the square of the side-length scale factor, so the area scale factor is 94\dfrac{9}{4}. The second triangle's area is 24×94=5424\times\dfrac{9}{4}=54.

Question 2

At a farmers market, seven sacks of rice cost the same as twenty-one sacks of flour. If one sack of flour costs $3, what is the price of one sack of rice?

  1. A$3
  2. B$9
  3. C$12
  4. D$15
Show answer

Answer: B — $9

Twenty-one sacks of flour cost 21×$3=$6321 \times \$3 = \$63, and that equals the cost of the 7 sacks of rice, so one sack of rice costs $63÷7=$9\$63 \div 7 = \$9. (A) just copies the given flour price, while (C) and (D) apply the wrong sack ratio (treating it as 4:14{:}1 or 5:15{:}1 instead of 3:13{:}1).

Question 3

5 identical machines produce 150 parts in 6 hours. At the same per-machine rate, how many parts will 7 machines produce in 6 hours?

  1. A35
  2. B152
  3. C210
  4. D1050
Show answer

Answer: C — 210

Each machine produces 150÷(5⋅6)=5150\div(5\cdot6)=5 parts per hour. Thus 7 machines produce 7(6)(5)=2107(6)(5)=210 parts.

Practice 105 Ratios and proportions; distance-rate-time questions in the app

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