Aimathic – Solve rate problems – Free Math Worksheets

Solve rate problems

Answer

a) $v=\frac{n}{m}$ pages per minute b) $T=\frac{x}{v}=\frac{xm}{n}$ minutes c) $40$ pages per minute d) $13.5$ minutes

5318306

A number line begins at $0$ and is divided into $20$ equal intervals. The endpoint is labeled “?”. Point $A$ is at the fourth tick mark from $0$. Find the value at “?” in each case. a) Point $A$ represents $0.8$. b) Point $A$ represents $0.08$. c) Point $A$ represents $\frac{2}{5}$.

Hints

- Compare the fourth interval with the total of $20$ intervals. - Determine how many times the distance from $0$ to $A$ fits in the whole scale. - Multiply the value at $A$ by that scale factor.

Solution

1. Point $A$ is $4$ of the $20$ equal intervals from $0$ to the endpoint. Thus, the whole scale is $20\div4=5$ times the value at $A$. 2. For a), the endpoint is $5\times0.8=4$. 3. For b), the endpoint is $5\times0.08=0.4$. 4. For c), the endpoint is $5\times\frac{2}{5}=2$.

Answer

a) $4$ b) $0.4$ c) $2$

The drain would empty the aquarium in $33$ minutes $20$ seconds.

5353536

A scale has tick marks numbered from left to right, beginning with tick $0$ at the far left. The labeled value $2$ is at tick $20$, and point $A$ is at tick $12$. For each possible value of point $A$, determine the tick number where $0$ must be located. a) $A=0.4$ b) $A=1.2$ c) $A=-2$

Hints

- Count the intervals between point $A$ and the tick labeled $2$. - For each case, divide the value difference by that number of intervals. - Use the interval value to determine how far and in which direction to move from $A$ to $0$.

Solution

1. There are $20-12=8$ intervals between point $A$ and the tick labeled $2$. 2. For a), the value change is $2-0.4=1.6$, so one interval represents $1.6\div8=0.2$. Moving from $0.4$ to $0$ requires $0.4\div0.2=2$ intervals left, so $0$ is at tick $12-2=10$. 3. For b), one interval represents $(2-1.2)\div8=0.1$. Moving from $1.2$ to $0$ requires $1.2\div0.1=12$ intervals left, so $0$ is at tick $0$. 4. For c), one interval represents $(2-(-2))\div8=0.5$. Moving from $-2$ to $0$ requires $2\div0.5=4$ intervals right, so $0$ is at tick $12+4=16$.

Answer

a) Tick $10$ b) Tick $0$ c) Tick $16$

5244076

A motorboat is traveling upstream at a constant speed relative to the water. As it passes under a bridge, it loses a floating boat fender without the driver noticing. The boat continues upstream for $15$ minutes, then turns around immediately and travels downstream at the same speed relative to the water. It catches the fender at a second bridge $0.75$ mile downstream from the first bridge. Find the river current''s speed.

Hints

- Think about the boat and fender from the point of view of the moving water. - How long does the boat take to return to the same parcel of water? - Find the fender's total drifting time. - Use rate equals distance divided by time.

Solution

1. View the motion relative to the moving water. In this frame, the fender stays in one place while the boat travels away from it for $15$ minutes. 2. Because the boat returns at the same speed relative to the water, it takes another $15$ minutes to reach the fender. 3. The fender therefore drifts for a total of $30$ minutes, or $0.5$ hour. 4. Let $r$ be the current's speed in miles per hour. The fender travels with the current, so $0.5r = 0.75$. 5. Divide by $0.5$: $r = 1.5$.

Answer

The river current flows at $1.5\,\text{mph}$.

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