Proportional Relationships and Slope: practice solutions, Grade 8

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Practice solutions Grade 8 : Proportional Relationships and Slope — Zyro the alien explorer of Planète Maths

Written solutions to the chapter problems. Check each step, then correct yourself.

1 Typing speed ★★★

$\dfrac{135}{3} = 45$ words per minute.
$y = 45x$, so in 8 minutes: $45 \times 8 = 360$ words.

Answer: 45 words per minute, and 360 words in 8 minutes.

2 Smoothie prices ★★★

Price per cup: $\dfrac{10}{4} = 2.50$ dollars, so $y = 2.5x$.

Cups $x$	1	2	5	8
Cost $y$ ($)	2.50	5.00	12.50	20.00

Check: $2.5 \times 8 = 20$.

3 Proportional or not? ★★★

The ratios $\dfrac{6}{2}, \dfrac{12}{4}, \dfrac{15}{5}$ all equal 3. Yes, proportional, with $y = 3x$.
The ratios are $\dfrac{3}{1} = 3$, $\dfrac{5}{2} = 2.5$, $\dfrac{7}{3} \approx 2.33$. They are not equal, so no, not proportional.

4 Reading a slope ★★★

From $A$ to $B$: run $= 6 - 2 = 4$, rise $= 9 - 3 = 6$.

$m = \dfrac{6}{4} = \dfrac{3}{2} = 1.5$.

Answer: the slope is 1.5.

5 Two points ★★★

$m = \dfrac{11 - 2}{4 - 1} = \dfrac{9}{3} = 3$.

Answer: the slope is 3.

6 Highway speed ★★★

$\dfrac{150}{2.5} = 60$ miles per hour.

$60 \times 1.609 \approx 96.5$, so about 96.5 kilometers per hour.

7 Plotting y = 2x ★★★

The pairs are $(0, 0), (1, 2), (2, 4), (3, 6)$.

The points line up on a line through the origin. Each step of 1 to the right raises $y$ by 2, so the slope is 2, equal to the coefficient of $x$.

8 A steep negative slope ★★★

$m = \dfrac{-8 - 4}{5 - (-3)} = \dfrac{-12}{8} = -\dfrac{3}{2}$.

The slope is negative, so the line falls from left to right: $-1.5$ units in $y$ for each unit in $x$.

9 Horizontal and vertical ★★★

$m = \dfrac{5 - 5}{7 - 2} = \dfrac{0}{5} = 0$: a horizontal line.
$m = \dfrac{6 - (-1)}{3 - 3} = \dfrac{7}{0}$: division by zero, so the slope is undefined (a vertical line).

10 Apple prices ★★★

Store A: $\dfrac{9}{5} = 1.80$ dollars per pound. Store B: 1.95 dollars per pound. Since $1.80 < 1.95$, Store A is cheaper.
Store A: $12 \times 1.80 = 21.60$. Store B: $12 \times 1.95 = 23.40$. Savings: $23.40 - 21.60 = 1.80$ dollars.

11 A draining tank ★★★

$m = \dfrac{36 - 60}{8 - 0} = \dfrac{-24}{8} = -3$. The tank loses 3 gallons every minute.
No. The line does not pass through the origin (it starts at 60), and the ratio $\dfrac{y}{x}$ is not constant (for example $\dfrac{36}{8} = 4.5$).
Draining 60 gallons at 3 gallons per minute takes $\dfrac{60}{3} = 20$ minutes.

12 A missing coordinate ★★★

Slope: $\dfrac{10}{4} = 2.5$, so $y = 2.5x$.

For $x = 6$: $k = 2.5 \times 6 = 15$.

Check with the slope: $\dfrac{15 - 10}{6 - 4} = \dfrac{5}{2} = 2.5$.

13 Equation from a table ★★★

$\dfrac{7}{2} = 3.5$, $\dfrac{17.5}{5} = 3.5$, $\dfrac{31.5}{9} = 3.5$. The ratio is constant.

Answer: $k = 3.5$ and $y = 3.5x$.

14 Running pace ★★★

40 minutes $= \dfrac{2}{3}$ hour, so $\dfrac{6}{2/3} = 6 \times \dfrac{3}{2} = 9$ kilometers per hour.

$\dfrac{9}{1.609} \approx 5.6$ miles per hour.

15 Similar triangles on a line ★★★

Small: $\dfrac{3}{4} = 0.75$. Large: $\dfrac{9}{12} = \dfrac{3}{4} = 0.75$. The triangles are similar (same angles, sides scaled by 3), so the ratio is the same.
$\text{rise} = 20 \times \dfrac{3}{4} = 15$.

16 Wheelchair ramp ★★★

$m = \dfrac{2.5}{30} = \dfrac{1}{12}$.
$\dfrac{1.5}{\text{run}} = \dfrac{1}{12}$, so run $= 1.5 \times 12 = 18$ feet.

17 Find the unknown ★★★

$\dfrac{13 - k}{6 - 2} = 2.5$, so $13 - k = 10$, which gives $k = 3$.

Check: $\dfrac{13 - 3}{4} = 2.5$.

18 Two sprinklers ★★★

Sprinkler 1: $\dfrac{8}{5} = 1.6$. Sprinkler 2: 1.7. Since $1.7 > 1.6$, Sprinkler 2 has the steeper line and the greater rate.
Sprinkler 1: $1.6 \times 20 = 32$ gallons. Sprinkler 2: $1.7 \times 20 = 34$ gallons.

19 Collinear points ★★★

Slope of $AB$: $\dfrac{4 - (-2)}{2 - (-1)} = \dfrac{6}{3} = 2$. Slope of $BC$: $\dfrac{10 - 4}{5 - 2} = \dfrac{6}{3} = 2$. The slopes match and $B$ is shared, so the three points are on one line, $y = 2x$.
Slope of $CD$: $\dfrac{15 - 10}{8 - 5} = \dfrac{5}{3} \neq 2$. Equivalently, $2 \times 8 = 16 \neq 15$. So $D$ is not on the line.