Unit rate as slope

1. The line starts at (0, 0), so this is a proportional relationship: 0 hours → \$0.
2. The marked point (4,\ 60) means 4 hours of work earns \$60.
3. Unit rate = dollars ÷ hours: \frac{60 \text{ dollars}}{4 \text{ hours}} = 15 \text{ dollars per hour}
4. This is also the slope of the line: m = \frac{60 - 0}{4 - 0} = 15.

Dana earns $15 per hour.

Check: At 8 hours the line reaches \$120, and 120 \div 8 = 15 ✓

Tip: In a proportional graph, the unit rate = slope = the y-value when x = 1. Here, at 1 hour the line is at \$15.

1. The equation has the form y = kx, a proportional relationship with constant k = 1.85.
2. k is the slope: when x increases by 1 pound, y increases by 1.85 dollars.
3. Test it: x = 1 \Rightarrow y = 1.85(1) = 1.85. One pound costs \$1.85.

So 1.85 is the price per pound of apples, in dollars.

Check: 2 pounds: 1.85 \times 2 = \$3.70, which is twice the cost of 1 pound — consistent with a constant price per pound ✓

Tip: Read the units of the slope as "y-units per x-unit": dollars per pound.

1. The line passes through (0, 0) and the marked point (3, 39): after 3 hours the cyclist has gone 39 miles.
2. Speed is miles per one hour, so divide miles by hours: \frac{39 \text{ miles}}{3 \text{ hours}} = 13 \text{ miles per hour}
3. This is also the slope of the line: m = \frac{39 - 0}{3 - 0} = 13.

The cyclist rides at 13 miles per hour.

Check: After 1 hour the line should be at 13 miles, and the first labeled grid line on the vertical axis is 13 — the line crosses it at 1 hour ✓

Tip: For a proportional graph (a line through the origin), the unit rate is \frac{y}{x} of any point on the line, and it equals the slope.

1. Unit price = price ÷ number of ounces.
2. 18-ounce box: \frac{\$4.32}{18 \text{ oz}} = \$0.24 \text{ per ounce}
3. 24-ounce box: \frac{\$5.52}{24 \text{ oz}} = \$0.23 \text{ per ounce}
4. \$0.23 < \$0.24, so the 24-ounce box costs less per ounce.

The 24-ounce box is the better buy at $0.23 per ounce.

Check: 0.23 \times 24 = 5.52 ✓ and 0.24 \times 18 = 4.32 ✓

Tip: "Better buy" means the lower unit price, not the lower total price. Always divide dollars by units so the answer is "dollars per ounce."

1. Printer A (graph): the line passes through (0, 0) and (5, 90). \text{rate}_A = \frac{90 \text{ pages}}{5 \text{ min}} = 18 \text{ pages per minute}
2. Printer B (equation y = 20x): the slope is 20, so it prints 20 pages per minute.
3. Compare: 20 - 18 = 2.

Printer B is faster by 2 pages per minute.

Check: In 5 minutes Printer B prints 20 \times 5 = 100 pages vs. Printer A's 90 pages: B prints 10 more in 5 minutes, which is 10 \div 5 = 2 more per minute ✓

Tip: To compare rates from different forms (graph vs equation vs table), convert each to the same unit rate first.

1. The line goes through (0, 0), so it is proportional: y = kx.
2. Find k (the unit rate) with the point (3, 2): k = \frac{2 \text{ in}}{3 \text{ h}} = \frac{2}{3} \text{ inch per hour}
3. So the equation is y = \frac{2}{3}x

The answer is y = \tfrac{2}{3}x.

Check: x = 9: y = \frac{2}{3}(9) = 6 — matches the point (9, 6) ✓

Tip: For a proportional graph, k = \frac{y}{x} for any point on the line (except the origin).

1. Ben's rate from the point (2.5,\ 15): \frac{15 \text{ mi}}{2.5 \text{ h}} = 6 \text{ mi/h}
2. Ana's rate is the slope of d = 6.5t: 6.5 mi/h.
3. Distances in 4 hours: \text{Ana: } 6.5 \times 4 = 26 \text{ mi} \qquad \text{Ben: } 6 \times 4 = 24 \text{ mi}
4. Difference: 26 - 24 = 2.

Ana runs 2 more miles.

Check: Ana gains 6.5 - 6 = 0.5 mile every hour; over 4 hours that is 0.5 \times 4 = 2 miles ✓

Tip: "How many more in 4 hours" = (difference in rates) × (time). Don't stop at the rate difference.

1. The equation has the form y = kx with k = 0.12. This is a proportional relationship.
2. If x = 1 (one copy), then y = 0.12 \times 1 = 0.12. So 1 copy costs $0.12.
3. Each additional copy adds another $0.12, so 0.12 is the unit rate: dollars per copy.
4. There is no added constant, so there is no setup fee — 0 copies cost $0.

0.12 means each copy costs $0.12.

Check: 100 copies cost 0.12 \times 100 = \$12, which is $0.12 for each of the 100 copies ✓

Tip: Read the units of k as "y-units per x-unit": here, dollars per copy. Say the sentence "___ dollars for each copy" to make sure it sounds right.

1. For a proportional relationship y = kx, the constant is k = \frac{y}{x}.
2. Compute it for each column: \frac{3}{2} = 1.5, \qquad \frac{6}{4} = 1.5, \qquad \frac{10.5}{7} = 1.5
3. The ratio is constant, so k = 1.5 and the equation is y = 1.5x

The equation is y = 1.5x.

Check: 1.5 \times 2 = 3 ✓, 1.5 \times 4 = 6 ✓, 1.5 \times 7 = 10.5 ✓

Tip: An equation that fits the first column only (like y = x + 1) is a common trap. Test every column before choosing.

1. Boxes per minute: \frac{45 \text{ boxes}}{15 \text{ min}} = 3 \text{ boxes per minute}
2. One hour = 60 minutes, so multiply by 60: 3 \times 60 = 180 \text{ boxes per hour}
3. Another way: 15 minutes is \frac{1}{4} of an hour, so an hour has 4 such periods: 45 \times 4 = 180.

The belt moves 180 boxes per hour.

Check: At 180 boxes per hour, a quarter hour (15 minutes) gives 180 \div 4 = 45 boxes ✓

Tip: When the question asks for a different time unit, convert at the end and ask: "Should the number get bigger or smaller?" An hour is longer than a minute, so more boxes per hour.

1. Store A's unit rate: \dfrac{3.78}{2} = 1.89 dollars per pound (check: 5.67 \div 3 = 1.89, 9.45 \div 5 = 1.89).
2. Store B's unit rate: in y = 1.95x, k = 1.95 dollars per pound.
3. Compare: 1.89 < 1.95, so Store A is cheaper.
4. Difference: 1.95 - 1.89 = 0.06 dollars per pound.

Store A is cheaper by $0.06 per pound.

Check: For 10 pounds: Store A 10 \times 1.89 = \$18.90, Store B 10 \times 1.95 = \$19.50. Difference \$0.60 for 10 pounds, which is $0.06 per pound ✓

Tip: To compare two representations, turn both into the same kind of number — the unit rate — before comparing.

1. Proportional means y = kx: the ratio \frac{y}{x} is identical for every pair (and the graph is a straight line through the origin).
2. Test each table:
   • x: 1, 2, 3; y: 2.5, 5, 7.5 → ratios 2.5, 2.5, 2.5 ✓ proportional.
   • x: 1, 2, 3; y: 3, 5, 7 → ratios 3, 2.5, 2.33 — not constant (this is y = 2x + 1).
   • x: 2, 4, 6; y: 4, 16, 36 → ratios 2, 4, 6 — not constant (this is y = x^2).
   • x: 1, 2, 4; y: 6, 12, 18 → ratios 6, 6, 4.5 — not constant.

The proportional table is x: 1, 2, 3 and y: 2.5, 5, 7.5, with k = 2.5.

Check: y = 2.5x: 2.5(1) = 2.5, 2.5(2) = 5, 2.5(3) = 7.5 ✓

Tip: A table can go up steadily and still not be proportional. Compute \frac{y}{x} for every pair — one mismatch rules it out.

1. Unit rate = miles ÷ hours: \frac{2}{3} \div \frac{1}{4}
2. Dividing by \frac{1}{4} is the same as multiplying by 4: \frac{2}{3} \times \frac{4}{1} = \frac{8}{3}
3. As a mixed number: \frac{8}{3} = 2\frac{2}{3} \approx 2.67.

The speed is \tfrac{8}{3} miles per hour (2\tfrac{2}{3} mph).

Check: In 1 hour there are four quarter-hours, and 4 \times \frac{2}{3} = \frac{8}{3} miles ✓

Tip: On the graph of distance vs time, this is the slope \frac{2/3}{1/4} = \frac{8}{3}. A "complex fraction" rate is still just top ÷ bottom.

1. Sugar per cup of flour = cups of sugar ÷ cups of flour: \frac{\tfrac{2}{3}}{\tfrac{1}{2}} = \frac{2}{3} \div \frac{1}{2}
2. Dividing by a fraction = multiplying by its reciprocal: \frac{2}{3} \times \frac{2}{1} = \frac{4}{3}
3. \tfrac{4}{3} = 1\tfrac{1}{3} cups of sugar per cup of flour.

The recipe uses \tfrac{4}{3} (or 1\tfrac{1}{3}) cups of sugar per cup of flour.

Check: One cup of flour is two half-cups. Each half-cup needs \tfrac{2}{3} cup of sugar, so 2 \times \tfrac{2}{3} = \tfrac{4}{3} ✓

Tip: "Per" means divide, and the unit after "per" goes on the bottom. With fractions, flip the bottom fraction and multiply.

1. Pump A (graph): through (0, 0) and (5, 60): \frac{60}{5} = 12 \text{ gallons per minute}
2. Pump B (words): \frac{100}{8} = 12.5 \text{ gallons per minute}
3. Time for 300 gallons = 300 ÷ rate: \text{A: } \frac{300}{12} = 25 \text{ min}, \qquad \text{B: } \frac{300}{12.5} = 24 \text{ min}
4. Pump B is faster: 25 - 24 = 1 minute less.

Pump B is faster, by 1 minute.

Check: 12.5 \times 24 = 300 ✓ and 12 \times 25 = 300 ✓

Tip: A difference in rates (0.5 gallon per minute) is not a difference in time. Reread what the question measures, then compute that quantity for each option.

1. Unit rate (height per hour): \frac{6 \text{ cm}}{15 \text{ h}} = 0.4 \text{ cm per hour}
2. Time for 10 cm = height ÷ rate: \frac{10}{0.4} = 25 \text{ hours}
3. Another way: hours per centimeter is \frac{15}{6} = 2.5, so 10 cm takes 10 \times 2.5 = 25 hours.

It will take 25 hours.

Check: 0.4 \times 25 = 10 cm ✓

Tip: Proportional situations scale by multiplying, not by adding. Going from 6 cm to 10 cm is not "+4 hours"; it is \frac{10}{6} times as long.

1. Water per cup of concentrate = water ÷ concentrate.
2. Use the last column (it has a whole number of cups): \frac{10\tfrac{1}{2}}{3} = \frac{10.5}{3} = 3.5
3. Check the first column: 1\tfrac{3}{4} \div \tfrac{1}{2} = 1.75 \div 0.5 = 3.5 ✓
4. Check the middle column: 5\tfrac{1}{4} \div 1\tfrac{1}{2} = 5.25 \div 1.5 = 3.5 ✓

The mixture uses 3.5 cups of water per cup of concentrate.

Check: 3.5 \times 1.5 = 5.25 = 5\tfrac{1}{4} ✓

Tip: Choose the easiest column to compute the unit rate (a whole-number input helps), then confirm with one other column.

1. Unit rate (slope) from (8, 6): k = \frac{6}{8} = 0.75 \text{ dollars per pound} So the line is y = 0.75x.
2. Test each choice:
   • (20, 15): 0.75 \times 20 = 15 ✓
   • (6, 8): 0.75 \times 6 = 4.5 \ne 8 ✗
   • (10, 8): 0.75 \times 10 = 7.5 \ne 8 ✗
   • (16, 14): 0.75 \times 16 = 12 \ne 14 ✗

The point (20,\ 15) is on the line.

Check: \frac{15}{20} = \frac{3}{4} = \frac{6}{8} — the same ratio, so the point is on the same proportional line ✓

Tip: Points on a proportional line keep the same ratio \frac{y}{x}. Adding the same amount to both coordinates (as in (10, 8) or (16, 14)) does not keep the ratio.

1. Unit price = price ÷ ounces.
2. 17 oz: \dfrac{6.29}{17} = \$0.37 per ounce.
3. 34 oz: \dfrac{11.90}{34} = \$0.35 per ounce.
4. 51 oz: \dfrac{18.36}{51} = \$0.36 per ounce.
5. Lowest: \$0.35 < \$0.36 < \$0.37.

The 34-ounce bottle has the lowest price, $0.35 per ounce.

Check: 0.35 \times 34 = 11.90 ✓, 0.36 \times 51 = 18.36 ✓, 0.37 \times 17 = 6.29 ✓

Tip: Stores don't always give the best deal on the largest size. Compute every unit price; it only takes one division each on the calculator.

1. Change miles to feet: 18 \text{ mi/h} \times 5{,}280 \text{ ft/mi} = 95{,}040 \text{ ft/h}
2. One hour = 60 \times 60 = 3{,}600 seconds. Change "per hour" to "per second" by dividing by 3,600: \frac{95{,}040}{3{,}600} = 26.4 \text{ ft/s}
3. As one expression: 18 \times \frac{5{,}280}{3{,}600} = 26.4

The speed is 26.4 feet per second.

Check: 26.4 \text{ ft/s} \times 3{,}600 \text{ s} = 95{,}040 ft in one hour, and 95{,}040 \div 5{,}280 = 18 miles ✓

Tip: Convert both units: the distance unit (miles → feet) and the time unit (hours → seconds). Writing the units in every step shows which ones cancel.