For a particular delivery, a courier travels by bicycle at a speed of $v$ miles per hour for a distance of $1.3$ miles. After making the delivery, the courier travels the same distance back, but travels $2$ miles per hour faster than on the way to the delivery. If the courier spent $0.2$ hours travelling to and from the delivery, which of the following equations could be used to determine the speed of the bicycle on the way to the delivery? $\newline$ Choose $1$ answer: $\newline$ (A) $0.2v^{2}-2.2v-2.6=0$ $\newline$ (B) $1.3v^{2}-2v-0.2=0$ $\newline$ (C) $0.2(v^{2}-3)+2.6=0$ $\newline$ (D) $3(0.2-v)(2.6-v)=0$

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Grade 7

Solve two-step equations: word problems

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Q. For a particular delivery, a courier travels by bicycle at a speed of

v

miles per hour for a distance of

1.3

miles. After making the delivery, the courier travels the same distance back, but travels

2

miles per hour faster than on the way to the delivery. If the courier spent

0.2

hours travelling to and from the delivery, which of the following equations could be used to determine the speed of the bicycle on the way to the delivery?

\newline

Choose

1

answer:

\newline

(A)

0.2v^{2}-2.2v-2.6=0

\newline

(B)

1.3v^{2}-2v-0.2=0

\newline

(C)

0.2(v^{2}-3)+2.6=0

\newline

(D)

3(0.2-v)(2.6-v)=0

Define Speed Variables: Let's denote the speed of the bicycle on the way to the delivery as $v$ (in miles per hour). The distance for one way is $1.3$ miles. The time taken to travel to the delivery at speed $v$ is the distance divided by the speed, which is $\frac{1.3}{v}$ hours.
Calculate Time to Delivery: On the way back, the courier travels $2$ miles per hour faster, so the speed is $v + 2$ miles per hour. The time taken to travel back is the same distance ( $1.3$ miles) divided by the new speed, which is rac{1.3}{v + 2} hours.
Calculate Time on Return Trip: The total time spent travelling to and from the delivery is the sum of the times for each leg of the trip, which is given as $0.2$ hours. So, we can write the equation as $\frac{1.3}{v} + \frac{1.3}{v + 2} = 0.2$ .
Formulate Total Time Equation: To find a common denominator and combine the terms on the left side of the equation, we multiply the first term by $(v + 2)/(v + 2)$ and the second term by $v/v$ . This gives us $(1.3(v + 2) + 1.3v) / (v(v + 2)) = 0.2$ .
Combine Terms and Simplify: Expanding the numerator, we get $(1.3v + 2.6 + 1.3v) / (v(v + 2)) = 0.2$ . Simplifying the numerator, we have $(2.6v + 2.6) / (v(v + 2)) = 0.2$ .
Clear Fractions: To clear the fraction, we multiply both sides of the equation by $v(v + 2)$ , which gives us $2.6v + 2.6 = 0.2v(v + 2)$ .
Expand and Rearrange Equation: Expanding the right side of the equation, we get $2.6v + 2.6 = 0.2v^2 + 0.4v$ .
Combine Like Terms: Rearranging the equation to set it to zero, we subtract $2.6v$ and $2.6$ from both sides to get $0 = 0.2v^2 + 0.4v - 2.6v - 2.6$ .
Final Equation Match: Simplifying the equation, we combine like terms to get $0 = 0.2v^2 - 2.2v - 2.6$ .
Final Equation Match: Simplifying the equation, we combine like terms to get $0 = 0.2v^2 - 2.2v - 2.6$ .This equation matches option (A), which is $0.2v^2 - 2.2v - 2.6 = 0$ .