SAT-49: Ratios, Rates, and Proportions

SAT-49: Ratios, Rates, and Proportions

Description: A ratio compares two quantities, a rate is a ratio with different units (like miles per hour), and a proportion sets two ratios equal. These power a huge share of SAT word problems.

Definitions

• Ratio: 3 : 5 or 3/5 — a comparison.
• Rate: a ratio of different units, e.g. 60 miles / 1 hour.
• Unit rate: a rate with denominator 1 (e.g. 60 mph, $3 per item). Proportion: two equal ratios, a/b = c/d. (Oʻzbekcha: nisbat ikki miqdorni taqqoslaydi; proporsiya ikkita nisbatni tenglashtiradi.) Solving a proportion Use cross-multiplication: if a/b = c/d then a·d = b·c. (Oʻzbekcha: proporsiyani krestga koʻpaytirib yechamiz: a·d = b·c.) Worked Example If 3 apples cost $1.20, how much do 7 apples cost?
  • Set up: 3/1.20 = 7/x.
  • Cross-multiply: 3x = 7(1.20) = 8.40.
  • x = 8.40/3 = $2.80. (Oʻzbekcha: bir xil birliklar mos joyda boʻlishi kerak — olma olmaga, narx narxga.) Worked Example (ratio split) $60 is split between two people in the ratio 2 : 3. How much does each get?
    • Total parts = 2 + 3 = 5. One part = 60/5 = 12.
    • Shares: 2·12 = $24 and 3·12 = $36.

    Practice

    A car travels 150 miles in 3 hours. At the same rate, how far in 5 hours?

    Show answer

    Unit rate = 150/3 = 50 mph. In 5 hours: 50 × 5 = 250 miles.

    Key words — Kalit soʻzlar

    • Ratio — nisbat
    • Rate — tezlik (sur'at)
    • Unit rate — birlik tezlik
    • Proportion — proporsiya
    • Cross-multiply — krestga koʻpaytirish
    • Quantity — miqdor
    • Per — har biriga
    • Share / Part — ulush / qism
    • Total — jami

    Summary

    • Ratio = comparison; rate = ratio of different units; unit rate has denominator 1.
    • Solve proportions by cross-multiplying (a/b = c/d → a·d = b·c).
    • Ratio split: add the parts, find one part, multiply out.