Algebra 1
Proportional Reasoning (the bridge to linearity)
Fresh, parallel-form problems with full worked solutions — more reps for the skills in this unit, kept separate from the textbook's own problems.
Lesson 3.1: Ratios and rates
Write the ratio of 10 red marbles to 14 blue marbles in lowest terms.
Worked solutionTry it first, then open.
A ratio reduces exactly like a fraction: divide both parts by their common factor. 10 and 14 share a factor of 2, so 10:14 = (10 ÷ 2):(14 ÷ 2) = 5:7. The value didn't change — 10:14 and 5:7 name the same comparison with smaller numbers. Order matters: red is named first, so red goes first (5:7), not 7:5.
Answer: 5:7
A train travels 210 miles in 6 hours. Find the unit rate in miles per hour (mph).
Worked solutionTry it first, then open.
A unit rate has a 1 in the second quantity — "per 1 hour." Divide to make the denominator 1: 210 mi / 6 hr = (210 ÷ 6)/(6 ÷ 6) mi/hr = 35 mi / 1 hr = 35 mph. Hold onto "35 miles per 1 hour" — in Unit 5 that same number is the slope of the distance-vs-time line.
Answer: 35
\$24 buys 8 notebooks. What does each notebook cost?
Worked solutionTry it first, then open.
Unit price means cost per 1 notebook, so the bottom must be 1 — divide by 8: \$24 / 8 notebooks = (\$24 ÷ 8)/(8 ÷ 8) = \$3 / 1 notebook = \$3 per notebook. Each notebook costs \$3.
Answer: 3
A smoothie recipe uses 8 cups of fruit for 20 servings. How many cups of fruit per serving? Give a reduced fraction.
Worked solutionTry it first, then open.
Write the rate as a fraction and reduce: 8 cups / 20 servings = (8 ÷ 4)/(20 ÷ 4) = 2/5 cup per serving. Fewer than one cup per serving is a perfectly real fraction — it's a rate (two unlike units), not a bare number, so keep "cup per serving" attached: ⅖ cup per serving.
Answer: 2/5
Lesson 3.2: Proportions & cross-multiplication
Solve the proportion x/5 = 12/20.
Worked solutionTry it first, then open.
Cross-multiplication is just "clear both denominators": multiply both sides by 5 and by 20. That gives 20·x = 5·12, i.e. 20x = 60, so x = 3. Check: 3/5 = 12/20 (both equal 0.6).
Answer: 3
Solve the proportion 4/x = 10/15. (Here x ≠ 0, since 4/x must be defined.)
Worked solutionTry it first, then open.
Cross-multiply (clear both denominators): 4·15 = 10·x, so 60 = 10x, giving x = 6. Check: 4/6 = 10/15 (both reduce to ⅔). Note x ≠ 0 was required for 4/x to make sense — and 6 ≠ 0, so we're fine.
Answer: 6
On a map, 1 inch represents 40 miles. How many miles does 4.5 inches represent? Set up a proportion, then solve.
Worked solutionTry it first, then open.
Keep the same units on top on both sides — inches over miles on each side: (1 inch)/(40 mi) = (4.5 inches)/(x mi). Cross-multiply: 1·x = 40·4.5, so x = 180 miles. (Sanity check: 4.5 inches is 4.5 times the 1-inch unit, and 4.5 × 40 = 180.)
Answer: 180
6 workers paint a fence in 10 days. Is this direct or inverse? Set it up and find how long 4 workers take at the same total work.
Worked solutionTry it first, then open.
Inverse — fewer workers means MORE days, so the quantities move in opposite directions. Don't cross-multiply 6/4 = 10/t; instead set the PRODUCTS equal. The total work is fixed: workers × days = 6 × 10 = 60 worker-days. So 4 × t = 60, giving t = 15 days. (The direct setup 6/4 = 10/t would wrongly give t = 6.67 days — fewer than 10, which can't be right when you have fewer workers.)
Answer: inverse: 6*10 = 4*t, t = 15 days
Lesson 3.3: Percents
What is 30% of 90?
Worked solutionTry it first, then open.
This is the "find the part" type: part = percent × whole, with the percent as a decimal. 30% = 0.30, so 0.30 × 90 = 27. (Check against the skeleton part/whole = percent/100: 27/90 = 30/100.)
Answer: 27
21 is what percent of 60?
Worked solutionTry it first, then open.
This is the "find the percent" type. part/whole = percent/100, so 21/60 = 0.35 = 35%. Read the sentence rather than reaching for "of means multiply" — here the percent is the missing slot, so you divide the part by the whole: 21 ÷ 60 = 0.35 = 35%.
Answer: 35
18 is 40% of what number?
Worked solutionTry it first, then open.
This is the "find the whole" type. part = percent × whole, so 0.40 · w = 18. Divide both sides by 0.40 (a Unit 2 one-step solve): w = 18 / 0.40 = 45. (Check: 0.40 × 45 = 18.)
Answer: 45
A subscription rises from \$60 to \$75 per year. What is the percent increase?
Worked solutionTry it first, then open.
Percent change is measured against the ORIGINAL (old) value — the denominator is always where you started. ((new − old)/old) × 100% = ((75 − 60)/60) × 100% = (15/60) × 100% = 25%. The result is positive, so it's a 25% increase. (Beware the wrong base: dividing 15 by the new \$75 would give 20% — that answers a different question.)
Answer: 25
Mixed review
Problems that mix skills from across the unit — good for spacing earlier work back in.
A delivery van covers 300 miles in 5 hours. Find the unit rate (mph), and note: this is the same number that becomes the slope of its distance-vs-time line.
Worked solutionTry it first, then open.
Divide to land on "per 1 hour": 300 mi / 5 hr = (300 ÷ 5)/(5 ÷ 5) = 60 mi / 1 hr = 60 mph. This unit rate — 60 miles per 1 hour — is a constant rate of change, the literal seed of slope in Unit 5.
Answer: 60
Solve the proportion 3/8 = x/24. Before solving, will x be bigger or smaller than 3?
Worked solutionTry it first, then open.
Since 24 is 3 times 8, x must be 3 times 3 — so x will be BIGGER than 3 (expect 9). Cross-multiply (clear both denominators): 3·24 = 8·x, so 72 = 8x, giving x = 9. Check: 3/8 = 9/24 (both reduce to ⅜).
Answer: 9
24 is 60% of what number?
Worked solutionTry it first, then open.
"Find the whole" type: part = percent × whole, so 0.60 · w = 24. Divide both sides by 0.60: w = 24 / 0.60 = 40. (Check: 0.60 × 40 = 24. Or as a proportion, 24/w = 60/100 ⇒ 60w = 2400 ⇒ w = 40.)
Answer: 40
8 identical machines fill an order in 6 hours. Is this direct or inverse? How long would 3 of those machines take on the same order?
Worked solutionTry it first, then open.
Inverse — fewer machines means MORE time, so they move in opposite directions; set the PRODUCTS equal, not the ratios. Total work = machines × hours = 8 × 6 = 48 machine-hours (fixed). So 3 × t = 48, giving t = 16 hours. (The direct trap 8/3 = 6/t would wrongly give t = 2.25 hours — less time with fewer machines, which is impossible.)
Answer: inverse: 8*6 = 3*t, t = 16 hours
A \$60 restaurant bill gets an 18% tip. What is the total? (Two steps: find the tip, then add.)
Worked solutionTry it first, then open.
Step 1 — find the part (the tip): 0.18 × 60 = 10.8, so the tip is \$10.80. Step 2 — add it to the bill: 60 + 10.80 = \$70.80. (Shortcut: a bill plus an 18% tip is 1.18 × 60 = \$70.80.)
Answer: 70.8