Level 2 · Enrichment

Reveal Math · Unit 3 · Supplemental

Ratios & Rates

Grade 6 · Standards 6.RP.A.1–3 — proportional reasoning, best buys & multi-step rates

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Challenge Problems

Directions: Solve and show your reasoning. For each "Explain" prompt, justify your answer in a complete sentence using unit rates or equivalent ratios.
  1. Store A sells 6 granola bars for $4.50; Store B sells 10 for $7.00. Which is the better buy? Real-world
    Explain: how do unit rates let you compare fairly?
  2. A printer prints 18 pages in 24 seconds. At this rate, how long to print 60 pages? Multi-step
  3. A map scale is 1 inch : 25 miles. Two cities are 3.5 inches apart. How far apart are they in real life? Real-world
  4. Paint is mixed 3 parts blue to 2 parts yellow. To make 30 cups of paint, how many cups of each color? Multi-step
    Explain: how did the total of 5 parts help you?
  5. One runner does 3 miles in 27 minutes; another does 5 miles in 40 minutes. Who is faster? Reasoning
    Explain using minutes-per-mile.
  6. A car uses 12 gallons to travel 384 miles. How far can it go on a full 16-gallon tank? Multi-step
  7. Write three different ratios equivalent to 8 : 12 that have not appeared before, including one in simplest form. Open-ended
  8. A juice mix is 4 parts water to 1 part concentrate. If you have 5 cups of concentrate, how much total juice can you make? Multi-step
  9. If 5 workers build a wall in 12 days, what assumptions let you say 10 workers finish in 6 days, and when might that fail? Reasoning
    Explain when the proportion does and does not hold.
  10. A recipe serving 4 uses 3 cups of rice. Rewrite it to serve 10. Then explain why you cannot just add 6 cups. Reasoning

Stretch Investigation

Real-world application: Find three grocery items sold in at least two package sizes (online or in a store ad). For each, compute the unit price and decide which size is the best value. Build a ratio table for one item, then write a recommendation explaining when the larger size is NOT actually the better buy.

Answer Key

  1. A: $4.50 ÷ 6 = $0.75/bar; B: $7.00 ÷ 10 = $0.70/bar. Store B is the better buy.
  2. 18 ÷ 24 = 0.75 pages/sec → 60 ÷ 0.75 = 80 seconds.
  3. 3.5 × 25 = 87.5 miles.
  4. 5 parts total → 30 ÷ 5 = 6 per part; blue 3 × 6 = 18 cups, yellow 2 × 6 = 12 cups.
  5. 9 min/mile vs 8 min/mile; the second runner (40 ÷ 5 = 8) is faster.
  6. 384 ÷ 12 = 32 mpg → 32 × 16 = 512 miles.
  7. Answers vary, e.g., 2 : 3 (simplest), 16 : 24, 24 : 36.
  8. 4 + 1 = 5 parts per 1 concentrate → 5 × 5 = 25 cups total juice.
  9. Assumes each worker works at the same constant rate independently; fails if the job can't be split (e.g., space limits or setup time).
  10. 3 ÷ 4 = 0.75 cups/serving → 0.75 × 10 = 7.5 cups. Adding 6 assumes 1 cup per serving, which changes the ratio.
  11. Stretch: answers vary; full credit needs correct unit prices, a ratio table, and a justified recommendation.