Number and Operations>Representing Constant Rates of Change Including d = rt(TEKS.Math.7.4.A)
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Texas 7th Grade Math › Number and Operations>Representing Constant Rates of Change Including d = rt(TEKS.Math.7.4.A)
A train travels at a constant speed. At 2 hours it has gone 150 miles; at 5 hours it has gone 375 miles. Use $d = rt$.
What is the train's constant speed?
50 miles/hour
60 miles/hour
75 miles/hour
225 miles/hour
Explanation
A constant rate means the ratio distance/time stays the same. Compute $r = \frac{d}{t}$: $\frac{150}{2} = 75$ and $\frac{375}{5} = 75$. In $d = rt$, $r$ is 75, so the speed is 75 miles/hour.
The relationship between hours worked (x) and pay (y) is represented by the ordered pairs: (2, 26), (5, 65), (8, 104). What is the constant of proportionality $k$ in $y=kx$?
45670
5
13
8
Explanation
The constant of proportionality is the consistent ratio $k=\dfrac{y}{x}$. Using (2, 26): $26/2=13$; using (5, 65): $65/5=13$; using (8, 104): $104/8=13$. So $k=13$, the unit rate (slope) of dollars per hour.
A bakery packs 1,260 cookies in 7 hours. What is the unit rate in cookies per hour?
0.0056 cookies/hour
160 cookies/hour
180 cookies/hour
7 cookies/hour
Explanation
Divide total cookies by hours to get per 1 hour: $1260 \div 7 = 180$, so 180 cookies/hour. Unit rates are useful for comparing production speeds between different days or bakeries.
Consider these four representations of distance over time.
Which representation shows a constant rate of change?
Table: time (h) 1, 2, 3; distance (mi) 52, 104, 156
Table: time (h) 1, 2, 3; distance (mi) 50, 110, 165
Equation: $d = 120t^2$
Verbal: A jogger speeds up, covering 0.9 mile in the first 10 minutes and 1.1 miles in the next 10 minutes
Explanation
A constant rate has a consistent ratio $\frac{d}{t}$. Choice A: $\frac{52}{1}=52$, $\frac{104}{2}=52$, $\frac{156}{3}=52$ (constant). B changes (50, 55, 55). C is quadratic ($t^2$) so the rate varies with $t$. D describes speeding up, not a constant rate.
A school places 252 students into 9 buses. What is the unit rate in students per bus?
28 students/bus
0.036 bus/student
9 students/bus
27 students/bus
Explanation
Divide students by buses to get per 1 bus: $252 \div 9 = 28$, so 28 students/bus. Unit rates help compare how crowded different buses are.
When $x$-values are 4, 6, 8, the corresponding $y$-values are 12, 18, 24. What is the value of $k$ in $y=kx$?
3
45660
2
4
Explanation
In a proportional relationship, $k=\dfrac{y}{x}$. Compute: $12/4=3$, $18/6=3$, and $24/8=3$. The ratio $y/x$ is the same for all pairs, so $k=3$ (the unit rate/slope).
A tank is being filled. Points from its volume–time graph are $(0, 5)$, $(2, 11)$, and $(6, 23)$, where time is in minutes and volume is in gallons.
What is the constant rate of change of the tank's volume?
1.5 gallons/minute
2 gallons/minute
2.5 gallons/minute
3 gallons/minute
Explanation
Constant rate equals the slope $\frac{\Delta V}{\Delta t}$. From $(0,5)$ to $(2,11)$: $\frac{11-5}{2-0}=\frac{6}{2}=3$. From $(2,11)$ to $(6,23)$: $\frac{23-11}{6-2}=\frac{12}{4}=3$. The rate is 3 gallons/minute.
At a market, 3.5 pounds of apples cost \$8.75. How much do apples cost per pound?
\$8.75 per pound
\$2.45 per pound
\$0.40 per pound
\$2.50 per pound
Explanation
Divide cost by pounds to get per 1 pound: $8.75 \div 3.5 = 2.50$, so \$2.50 per pound. Unit price lets you compare costs across different bag sizes or stores.
A recipe relates batches baked ($x$) to cups of flour used ($y$) with these pairs: (1, 3), (3, 9), (4, 12). What is the constant of proportionality $k$ in $y=kx$?
12
2
45660
3
Explanation
The constant of proportionality is $k=\dfrac{y}{x}$. Using (1, 3): $3/1=3$; using (3, 9): $9/3=3$; using (4, 12): $12/4=3$. The ratio $y/x$ is consistent, so $k=3$ cups per batch (the unit rate/slope).
The relationship is proportional with $x$-values 2.5, 4, 6.5 and $y$-values 5, 8, 13. What is the value of $k$ in $y=kx$?
1
2
8/6.5
1
Explanation
Compute $k=\dfrac{y}{x}$ for any pair: $5/2.5=2$, $8/4=2$, $13/6.5=2$. Because $y/x$ is the same for all pairs, $k=2$ (the unit rate/slope). Distractors use $x/y$ or mix nonmatching values.