HSPT Math - How to do distance problems

On a map, one half of an inch represents forty miles of real distance.

It takes John 90 minutes to get from Kingsbury to Willoughby driving an average of miles per hour. How many inches apart, in terms of , are the two cities on the map?

$\frac{3}{160}R$

$\frac{3}{80}R$

$\frac{1}{120}R$

$\frac{1}{60}R$

Explanation

The distance in real miles between Kingsbury in Willoughby can be found by multiplying rate miles per hour by time 90 minutes, or one and a half hours:

$R \cdot 1 \frac{1}{2} = \frac{3}{2}R$

Let represent map distance between the cities, One half of an inch represents forty miles of real distance, so one inch represents twice this, ir eighty miles. The ratio that compares map distance and real distance is

$\frac{d}{ \frac{3}{2}R} = \frac{ 1}{80}$

$\frac{d}{ \frac{3}{2}R} \cdot \frac{3}{2}R = \frac{1}{80} \cdot \frac{3}{2}R$

$d = \frac{3}{160}R$

Mr. Thomas's car holds exactly 14 gallons of gasoline, and gets 24 miles per gallon. He gets into his car, which has a full gas tank, and drives miles. He then refills the car until the gas gauge reads "full" again. In terms of , how many gallons of gasoline did he put in his car?

$\frac{D}{24}$

$\frac{D}{14}$

$14 - \frac{D}{24}$

$24 - \frac{D}{14}$

Explanation

Mr. Thomas gets 24 miles per gallon, and has driven miles; divide distance by gallons used, and he has used $\frac{D}{24}$ gallons. Since he is refilling his car, he is putting $\frac{D}{24}$ gallons in his car.

Note that the amount of gasoline that the tank will hold is irrelevant to the problem.

Worker graphs

Six friends work for a company as maintenance staff.

Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.

Of the six, which one started out, realized he forgot his briefcase, went back for it, and went to work?

Mr. Idle

Mr. Gilliam

Mr. Palin

Mr. Chapman

Explanation

The distance this person was from his home increased steadily as he went to work, then decreased as he went back home for his briefcase, then increased again as he went on to work. This describes the graph for Mr. Idle.

Suppose a student ran a pace of eight minutes per mile at consistent pace. He arrived at the school in thirty minutes. How far is the school in miles?

$\frac{15}{4}$

$\frac{15}{2}$

$\frac{4}{15}$

$\frac{2}{15}$

$\frac{12}{5}$

Explanation

The consistent pacing tells us that this is a linear relationship between distance and the student's speed and time.

Write the equation for the distance travelled.

The speed can be rewritten as: $\frac{1 \textup{ mile}}{8 \textup{ min}}$

Substitute the speed and time.

$d=(\frac{1 \textup{ mile}}{8 \textup{ min}})(30 \textup{ min}) = \frac{30}{ 8}= \frac{15}{4} \textup{ miles}$

Find the distance from point to point .

$4\sqrt{2}$

$\sqrt2$

$\frac{5}{2}$

Explanation

Write the distance formula.

$d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Substitute the values of the points into the formula.

$d= \sqrt{(6-2)^2+(7-3)^2}$

$d= \sqrt{(4)^2+(4)^2} = \sqrt{16+16} = \sqrt{32}$

The square root of can be reduced because , a factor of , is a perfect square. .

Now we have $d=\sqrt{16*2}=4\sqrt{2}$

Worker graphs

Six friends work for a company as maintenance staff.

Of the six, who took the express train directly to work?

Mr. Cleese

Mr. Palin

Mr. Chapman

Mr. Jones

Explanation

The distance from home of the worker in question would have increased steadily without any interruption. The graph with the constantly increasing line is the one to choose; this graph belongs to Mr. Cleese.

Worker graphs

Six friends work for a company as maintenance staff.

Of the six, which one called in sick at 7:30 AM and stayed in bed?

Mr. Jones

Mr. Palin

Mr. Gilliam

Mr. Cleese

Explanation

Someone who called in sick would remain at constant zero distance from his home during the entire time. This would be represented by a horizontal line along the zero axis; this describes the graph for Mr. Jones.

Jason is driving across the country. For the first 3 hours, he travels 60 mph. For the next 2 hours he travels 72 mph. Assuming that he has not stopped, what is his average traveling speed in miles per hour?

$64.8\ mph$

$72\ mph$

$66.7\ mph$

$63.4\ mph$

Explanation

In the first three hours, he travels 180 miles.

$3\times60=180$

In the next two hours, he travels 144 miles.

$2\times 72=144$

for a total of 324 miles.

Divide by the total number of hours to obtain the average traveling speed.

$\frac{324}{5}=64.8\ mph$

Worker graphs

Six friends work for a company as maintenance staff.

Of the six, who stopped for some breakfast on the way to work?

Mr. Chapman

Mr. Idle

Mr. Jones

Mr. Palin

Explanation

We are looking for someone whose distance from home increased steadily for a while, then became constant (since the person had to have not been moving), then increased steadily again. This describes the graph for Mr. Chapman.

Joe drove an average of 45 miles per hour along a 60-mile stretch of highway, then an average of 60 miles per hour along a 30-mile stretch of highway. What was his average speed, to the nearest mile per hour?

$49 ; \textrm{mph}$

$55 ; \textrm{mph}$

$52 ; \textrm{mph}$

$46 ; \textrm{mph}$

$53 ; \textrm{mph}$

Explanation

At 45 mph, Joe drove 60 miles in $60 \div 45 = \frac{60}{45}= \frac{4}{3}$ hours.

At 60 mph, he drove 30 miles in $30 \div 60 = \frac{1}{2}$ hours.

He made the 90-mile trip in $\frac{4}{3} + \frac{1}{2} = \frac{8}{6} + \frac{3}{6} = \frac{11}{6}$ hours, so divide 90 by $\frac{11}{6}$ to get the average speed in mph:

$90 \div \frac{11}{6} = \frac{90}{1} \div \frac{11}{6} = \frac{90}{1} \cdot \frac{6}{11} = \frac{540} {11} \approx49$

How to do distance problems

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