Question 1

Jeff takes 5 hours longer to build a fence than it takes Bill. When they work together, it takes them 6 hours. How long would it take Bill to do the job alone?
A) 9 hours
B) 11 hours
C) 12 hours
D) 10 hours

Accepted Answer

When Jeff and Bill work together, their combined work rate is 1/6 of the job per hour (since it takes them 6 hours to complete the job together). Let's denote the time it takes for Bill to build the fence alone as $B$ hours. Therefore, Jeff takes $B + 5$ hours to do the job alone. The work rate of Bill is $1/B$ of the job per hour, and the work rate of Jeff is $1/(B + 5)$ of the job per hour. Adding their rates gives the equation $1/B + 1/(B + 5) = 1/6$. Solving this equation for $B$ gives $B = 10$ hours.

Question 2

Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 245 miles in the same time that Dana travels 215 miles. If Chuck's rate of travel is $6 \mathrm{mph}$ more than Dana's, and they travel the same length of time, at what speed does Chuck travel?
A) $56 \mathrm{mph}$
B) $42 \mathrm{mph}$
C) $49 \mathrm{mph}$
D) $43 \mathrm{mph}$

Accepted Answer

The answer of Chuck and Dana agree to meet in...

Question 3

The speed of a stream is $6 \mathrm{mph}$. If a boat travels 58 miles downstream in the same time that it takes to travel 29 miles upstream, what is the speed of the boat in still water?
A) $18 \mathrm{mph}$
B) $21 \mathrm{mph}$
C) $12 \mathrm{mph}$
D) $20 \mathrm{mph}$

Accepted Answer

Let the speed of the boat in still water be $x \mathrm{mph}$. Downstream speed is $x + 6$ and upstream speed is $x - 6$. The time taken for both journeys is the same, so $\frac{58}{x+6} = \frac{29}{x-6}$. Solving this equation gives $x = 18$.

Question 4

A plane flies 400 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is $25 \mathrm{mph}$, what is the speed of the plane in calm air?
A) $400 \mathrm{mph}$
B) $365 \mathrm{mph}$
C) $380 \mathrm{mph}$
D) $375 \mathrm{mph}$

Accepted Answer

The speed of the plane in calm air can be found by setting up equations based on the distance formula: distance = speed × time. Let $p$ be the speed of the plane in calm air. With the wind, the plane's speed is $p + 25$, and against the wind, it's $p - 25$. Since the times are equal, we have $400 / (p + 25) = 350 / (p - 25)$. Solving this equation for $p$ gives $p = 375 \mathrm{mph}$.

Question 5

Tom Quig traveled 220 miles east of St. Louis. For most of the trip he averaged $60 \mathrm{mph}$, but for one period of time he was slowed to $10 \mathrm{mph}$ due to a major accident. If the total travel time was 7 hours, for how many miles did he drive at the reduced speed?
A) 35 miles
B) 50 miles
C) 60 miles
D) 40 miles

Accepted Answer

Let $x$ be the distance traveled at $10 \mathrm{mph}$. Then, the distance traveled at $60 \mathrm{mph}$ is $220 - x$ miles. The time spent traveling at $10 \mathrm{mph}$ is $\frac{x}{10}$ hours, and the time spent traveling at $60 \mathrm{mph}$ is $\frac{220 - x}{60}$ hours. The total time is 7 hours, so $\frac{x}{10} + \frac{220 - x}{60} = 7$. Solving this equation for $x$ gives $x = 40$ miles.

Question 6

A boat can go 180 miles downstream in the same time it can go 120 miles upstream. The speed of the current is 4 miles per hour. Find the speed of the boat in still water.
A) $16 \mathrm{mph}$
B) $24 \mathrm{mph}$
C) $30 \mathrm{mph}$
D) $20 \mathrm{mph}$

Accepted Answer

The speed of the boat in still water can be found using the formula $v = \frac{d}{t}$, where $v$ is the speed, $d$ is the distance, and $t$ is the time. Let $x$ be the speed of the boat in still water. The speed downstream is $x + 4$ mph, and the speed upstream is $x - 4$ mph. Since the times are equal, we have $\frac{180}{x + 4} = \frac{120}{x - 4}$. Solving this equation for $x$ gives $x = 20$ mph.

Question 7

A man rode a bicycle for 12 miles and hiked an additional 8 miles. The total time for the trip was 5 hours. If his bicycling rate was 10 miles per hour faster than his hiking rate, then what was his hiking rate?
A) $2 \mathrm{mph}$
B) $4 \mathrm{mph}$
C) $1 \mathrm{mph}$
D) $3 \mathrm{mph}$

Accepted Answer

Let the hiking rate be $x$ mph. Then the bicycling rate is $x+10$ mph.Time taken for bicycling = $\frac{12}{x+10}$ hoursTime taken for hiking = $\frac{8}{x}$ hoursTotal time = $\frac{12}{x+10}+\frac{8}{x}=5$$\Rightarrow \frac{12x+80+8x+100}{x(x+10)}=5$$\Rightarrow 20x+180=5x(x+10)$$\Rightarrow 20x+180=5x^2+50x$$\Rightarrow 5x^2+30x-180=0$$\Rightarrow x^2+6x-36=0$$\Rightarrow (x+12)(x-3)=0$$\Rightarrow x=-12,3$Since the rate cannot be negative, the hiking rate is $3$ mph.

Question 8

A boat can go $90 \mathrm{mph}$ in still water. It takes as long to go 630 miles upstream as it does to go downstream 720 miles. How fast is the current?
A) $1 \mathrm{mph}$
B) $9 \mathrm{mph}$
C) $5 \mathrm{mph}$
D) $6 \mathrm{mph}$

Accepted Answer

The speed of the current can be found by setting the time to travel upstream equal to the time to travel downstream and solving for the current's speed. Let $c$ be the current's speed. The boat's effective speed upstream is $90 - c$ and downstream is $90 + c$. Since time = distance/speed, we have $\frac{630}{90-c} = \frac{720}{90+c}$. Solving this equation for $c$ gives $c = 6 \mathrm{mph}$.

Question 9

$y$ varies directly as $x$. If $y=63$ when $x=9$, find $y$ when $x=3$.
A) 81
B) 49
C) 21
D) 9

Accepted Answer

Since $y$ varies directly as $x$, we can write $y = kx$ where $k$ is the constant of variation. Given $y=63$ when $x=9$, we find $k = 63 / 9 = 7$. Thus, when $x=3$, $y = 7 \times 3 = 21$.

Question 10

$y$ varies directly as $x$. If $y=24$ when $x=8$, find $y$ when $x=16$.
A) 48
B) 12
C) 80
D) 192

Accepted Answer

Since $y$ varies directly as $x$, the ratio $y/x$ is constant. Given $y=24$ when $x=8$, the constant is $24/8=3$. For $x=16$, $y=3*16=48$.

Question 11

$y$ varies directly as $x$. If $y=\frac{17}{2}$ when $x=17$, find $y$ when $x=16$.
A) $\frac{289}{32}$
B) 8
C) $\frac{17}{16}$
D) $\frac{1}{2}$

Accepted Answer

Since $y$ varies directly as $x$, the relationship between $y$ and $x$ can be described by the equation $y = kx$, where $k$ is the constant of variation. Given $y = \frac{17}{2}$ when $x = 17$, we can find $k$ by substituting these values into the equation, resulting in $k = \frac{y}{x} = \frac{\frac{17}{2}}{17} = \frac{1}{2}$. To find $y$ when $x = 16$, substitute $x = 16$ and $k = \frac{1}{2}$ into the equation $y = kx$, yielding $y = \frac{1}{2} \times 16 = 8$.

Question 12

$y$ varies directly as the square of $x$. If $y=72$ when $x=3$, find $y$ when $x$ is 9 .
A) 648
B) 24
C) 216
D) 0
E) 27

Accepted Answer

The relationship between $y$ and $x$ is given by $y = kx^2$, where $k$ is the constant of variation. Given $y=72$ when $x=3$, we find $k = 72 / 3^2 = 8$. For $x=9$, $y = 8 \times 9^2 = 648$.

Question 13

$y$ varies inversely as $x$. If $y=6$ when $x=9$, find $y$ when $x=27$.
A) 81
B) 18
C) 2
D) 3

Accepted Answer

When $y$ varies inversely as $x$, the relationship can be described by the equation $y = \frac{k}{x}$, where $k$ is a constant. Given $y=6$ when $x=9$, we can find $k$ by substituting these values into the equation, giving $6 = \frac{k}{9}$, so $k = 54$. To find $y$ when $x=27$, substitute $x=27$ into the equation with $k=54$, giving $y = \frac{54}{27} = 2$.

Question 14

$y$ varies inversely as $x$. If $y=48$ when $x=6$, find $y$ when $x=8$.
A) 36
B) 6
C) 64
D) 18

Accepted Answer

Since $y$ varies inversely as $x$, we use the formula $y = k/x$, where $k$ is a constant. Given $y=48$ when $x=6$, we find $k = 48*6 = 288$. For $x=8$, $y = 288/8 = 36$.

Question 15

$y$ varies inversely as $x$. If $y=\frac{5}{2}$ when $x=5$, find $y$ when $x=6$.
A) $\frac{1}{3}$

B) $\frac{25}{12}$

C) $\frac{5}{6}$

D) 3

Accepted Answer

The relationship between $y$ and $x$ in inverse variation is given by $y = \frac{k}{x}$, where $k$ is a constant. Given $y = \frac{5}{2}$ when $x = 5$, we find $k = \frac{5}{2} \times 5 = \frac{25}{2}$. To find $y$ when $x = 6$, we use $y = \frac{\frac{25}{2}}{6} = \frac{25}{12}$.

Question 16

If $y$ varies inversely as the square of $x$. If $y=3$ when $x=24$, find $y$ when $x=6$.
A) 48
B) 108
C) 36
D) 4

Accepted Answer

The relationship between $y$ and $x$ is given by $y = k/x^2$, where $k$ is a constant. When $y=3$ and $x=24$, we can find $k$ by substituting these values into the equation, giving $k = 3 \times 24^2 = 3 \times 576 = 1728$. When $x=6$, $y = 1728/6^2 = 1728/36 = 48$.

Question 17

$\mathrm{y}$ varies jointly as $\mathrm{x}$ and $\mathrm{z}$. If $\mathrm{y}=882$ when $\mathrm{x}=7$ and $\mathrm{z}=18$, find $\mathrm{y}$ when $\mathrm{x}=6$ and $\mathrm{z}=3$.
A) 126
B) 111,132
C) 49
D) 162

Accepted Answer

The relationship is $y = kxz$, where $k$ is the constant of variation. First, find $k$ using $y=882$, $x=7$, and $z=18$. Then, use $k$ to find $y$ when $x=6$ and $z=3$.

Question 18

The number of patients Dr. Wong can see varies directly as the time it takes her to see them. If she can see 9 patients in 2 hours, how long would it take her to see 90 patients?
A) 405 hours
B) 18 hours
C) 19 hours
D) 20 hours

Accepted Answer

The answer of The number of patients Dr. Wong can...

Question 19

The time $T$ necessary to make an enlargement of a photo negative varies directly as the area $A$ of the enlargement. If 245 seconds are required to make a 5 - by- 7 in. enlargement, find the time required for a 7 - by- 10 in. enlargement.
A) 560 seconds
B) 630 seconds
C) 420 seconds
D) 490 seconds

Accepted Answer

The area of the 5-by-7 enlargement is $35$ square inches, and the area of the 7-by-10 enlargement is $70$ square inches. Since time varies directly with area, the ratio of the times is equal to the ratio of the areas: $\frac{T_2}{245} = \frac{70}{35}$. Solving for $T_2$, we get $T_2 = 490$ seconds.

Question 20

The number $G$ of gears a machine can make varies directly as the time $T$ it operates. If it can make 7665 gears in 4 hours, how many gears can it make in 2 hours?
A) 1 gears
B) 7671 gears
C) 1916 gears
D) 3833 gears

Accepted Answer

Since the number of gears $G$ varies directly with time $T$, we have $G = kT$ for some constant $k$. Given $G = 7665$ when $T = 4$, we find $k = \frac{7665}{4} = 1916.25$. For $T = 2$, $G = 1916.25 \times 2 = 3833$.

Quiz 6: Rational Expressions and Equations

Jeff takes 5 hours longer to build a fence than it takes Bill. When they work together, it takes them 6 hours. How long would it take Bill to do the job alone?

Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 245 miles in the same time that Dana travels 215 miles. If Chuck's rate of travel is $6 \mathrm{mph}$ more than Dana's, and they travel the same length of time, at what speed does Chuck travel?

The speed of a stream is $6 \mathrm{mph}$ . If a boat travels 58 miles downstream in the same time that it takes to travel 29 miles upstream, what is the speed of the boat in still water?

A plane flies 400 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is $25 \mathrm{mph}$ , what is the speed of the plane in calm air?

Tom Quig traveled 220 miles east of St. Louis. For most of the trip he averaged $60 \mathrm{mph}$ , but for one period of time he was slowed to $10 \mathrm{mph}$ due to a major accident. If the total travel time was 7 hours, for how many miles did he drive at the reduced speed?

A boat can go 180 miles downstream in the same time it can go 120 miles upstream. The speed of the current is 4 miles per hour. Find the speed of the boat in still water.

A man rode a bicycle for 12 miles and hiked an additional 8 miles. The total time for the trip was 5 hours. If his bicycling rate was 10 miles per hour faster than his hiking rate, then what was his hiking rate?

A boat can go $90 \mathrm{mph}$ in still water. It takes as long to go 630 miles upstream as it does to go downstream 720 miles. How fast is the current?

$y$ varies directly as $x$ . If $y=63$ when $x=9$ , find $y$ when $x=3$ .

$y$ varies directly as $x$ . If $y=24$ when $x=8$ , find $y$ when $x=16$ .

$y$ varies directly as $x$ . If $y=\frac{17}{2}$ when $x=17$ , find $y$ when $x=16$ .

$y$ varies directly as the square of $x$ . If $y=72$ when $x=3$ , find $y$ when $x$ is 9 .

$y$ varies inversely as $x$ . If $y=6$ when $x=9$ , find $y$ when $x=27$ .

$y$ varies inversely as $x$ . If $y=48$ when $x=6$ , find $y$ when $x=8$ .

$y$ varies inversely as $x$ . If $y=\frac{5}{2}$ when $x=5$ , find $y$ when $x=6$ .

If $y$ varies inversely as the square of $x$ . If $y=3$ when $x=24$ , find $y$ when $x=6$ .

$\mathrm{y}$ varies jointly as $\mathrm{x}$ and $\mathrm{z}$ . If $\mathrm{y}=882$ when $\mathrm{x}=7$ and $\mathrm{z}=18$ , find $\mathrm{y}$ when $\mathrm{x}=6$ and $\mathrm{z}=3$ .

The number of patients Dr. Wong can see varies directly as the time it takes her to see them. If she can see 9 patients in 2 hours, how long would it take her to see 90 patients?

The time $T$ necessary to make an enlargement of a photo negative varies directly as the area $A$ of the enlargement. If 245 seconds are required to make a 5 - by- 7 in. enlargement, find the time required for a 7 - by- 10 in. enlargement.

The number $G$ of gears a machine can make varies directly as the time $T$ it operates. If it can make 7665 gears in 4 hours, how many gears can it make in 2 hours?

Quiz 6: Rational Expressions and Equations

Jeff takes 5 hours longer to build a fence than it takes Bill. When they work together, it takes them 6 hours. How long would it take Bill to do the job alone?

Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 245 miles in the same time that Dana travels 215 miles. If Chuck's rate of travel is 6mph6 \mathrm{mph}6mph more than Dana's, and they travel the same length of time, at what speed does Chuck travel?

The speed of a stream is 6mph6 \mathrm{mph}6mph . If a boat travels 58 miles downstream in the same time that it takes to travel 29 miles upstream, what is the speed of the boat in still water?

A plane flies 400 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is 25mph25 \mathrm{mph}25mph , what is the speed of the plane in calm air?

Tom Quig traveled 220 miles east of St. Louis. For most of the trip he averaged 60mph60 \mathrm{mph}60mph , but for one period of time he was slowed to 10mph10 \mathrm{mph}10mph due to a major accident. If the total travel time was 7 hours, for how many miles did he drive at the reduced speed?

A boat can go 180 miles downstream in the same time it can go 120 miles upstream. The speed of the current is 4 miles per hour. Find the speed of the boat in still water.

A man rode a bicycle for 12 miles and hiked an additional 8 miles. The total time for the trip was 5 hours. If his bicycling rate was 10 miles per hour faster than his hiking rate, then what was his hiking rate?

A boat can go 90mph90 \mathrm{mph}90mph in still water. It takes as long to go 630 miles upstream as it does to go downstream 720 miles. How fast is the current?

yyy varies directly as xxx . If y=63y=63y=63 when x=9x=9x=9 , find yyy when x=3x=3x=3 .

yyy varies directly as xxx . If y=24y=24y=24 when x=8x=8x=8 , find yyy when x=16x=16x=16 .

yyy varies directly as xxx . If y=172y=\frac{17}{2}y=217​ when x=17x=17x=17 , find yyy when x=16x=16x=16 .

yyy varies directly as the square of xxx . If y=72y=72y=72 when x=3x=3x=3 , find yyy when xxx is 9 .

yyy varies inversely as xxx . If y=6y=6y=6 when x=9x=9x=9 , find yyy when x=27x=27x=27 .

yyy varies inversely as xxx . If y=48y=48y=48 when x=6x=6x=6 , find yyy when x=8x=8x=8 .

yyy varies inversely as xxx . If y=52y=\frac{5}{2}y=25​ when x=5x=5x=5 , find yyy when x=6x=6x=6 .

If yyy varies inversely as the square of xxx . If y=3y=3y=3 when x=24x=24x=24 , find yyy when x=6x=6x=6 .

y\mathrm{y}y varies jointly as x\mathrm{x}x and z\mathrm{z}z . If y=882\mathrm{y}=882y=882 when x=7\mathrm{x}=7x=7 and z=18\mathrm{z}=18z=18 , find y\mathrm{y}y when x=6\mathrm{x}=6x=6 and z=3\mathrm{z}=3z=3 .