Question 1

An inlet pipe can fill a tank in 15 minutes. A drain can empty the tank in 20 minutes. If the tank is empty and both the pipe and drain are open, how long will it take before the tank overflows?  __________ minutes

Accepted Answer

The answer of An inlet pipe can fill a tank...

Question 2

Kent drives his Mazda 250 miles in the same time that Dave drives his Nissan 225 miles. If Kent averages 5 miles per hour faster than Dave, find their rates.

Accepted Answer

Let's call the speeds of Kent and Dave "K" and "D" respectively. We know that Kent drove 250 miles and Dave drove 225 miles. We also know that they took the same amount of time. Therefore:

250/K = 225/D

We also know that K is 5 mph faster than D, so:

K = D + 5

Substitute D + 5 for K in the first equation:

250/(D+5) = 225/D

Solve for D:

D = 45 mph

Then, we can find K:

K = D + 5 = 50 mph

Therefore, the answer is B, with Dave's rate being 45 mph and Kent's rate being 50 mph.

Question 3

It takes Amy twice as long to deliver papers as it does Nancy. How long, in minutes, would it take each girl to deliver the papers by herself if they can deliver the papers together in 20 minutes?

Accepted Answer

If Nancy takes x minutes, Amy takes 2x minutes. Together, their rate is 1/20 papers per minute. Solving 1/x + 1/(2x) = 1/20 gives x = 30, so Nancy takes 30 minutes and Amy takes 60 minutes.

Question 4

Solve each equation for the variable y . $\frac { y - b } { x } = m$

Accepted Answer

The answer of Solve each equation for the variable y...

Question 5

Solve the equation. $\frac { x } { x - 3 } - \frac { 4 } { x + 7 } = \frac { 52 } { x ^ { 2 } + 4 x - 21 }$

Accepted Answer

To solve the equation, first, we need to find the common denominator of the left side of the equation:
$$\frac{x(x+7)}{(x-3)(x+7)}-\frac{4(x-3)}{(x-3)(x+7)}=\frac{52}{x^2+4x-21}$$
Simplifying,
$$\frac{x(x+7)-4(x-3)}{(x-3)(x+7)}=\frac{52}{(x-3)(x+7)}$$
$$\frac{x^2+3x-28}{(x-3)(x+7)}=\frac{52}{(x-3)(x+7)}$$
$$(x^2+3x-28)-52=0$$
$$(x-5)(x+8)=0$$
Thus, the solutions are $x=5$ and $x=-8$, so the answer is C.

Question 6

An inlet pipe can fill a tank in 10 minutes. A drain can empty the tank in 15 minutes. If the tank is empty and both the pipe and drain are open, how long will it take before the tank overflows?

Accepted Answer

The answer of An inlet pipe can fill a tank...

Question 7

If two inlet pipes are both open, they can fill a pool in 2 hours and 24 minutes. One of the pipes can fill the pool by itself in 6 hours. How long, in hours, would it take the other pipe to fill the pool by itself? __________ hours

Accepted Answer

The answer of If two inlet pipes are both open,...

Question 8

Solve the equation. $\frac { x } { 5 x ^ { 2 } + 6 x } - \frac { x } { 5 x ^ { 2 } + 11 x + 6 } = \frac { 4 } { x ^ { 2 } + x }$

Accepted Answer

The equation can be solved by finding a common denominator and simplifying. After simplification, the solution is $x = - \frac { 24 } { 19 }$.

Question 9

To travel 38 miles, it takes Sue, riding a moped, 2 hours less time than it takes Doreen to travel 48 miles riding a bicycle. Sue travels 7 miles per hour faster than Doreen. Find the times and rates of both girls.

Accepted Answer

Let's denote Doreen's speed as $d$ mph. Therefore, Sue's speed is $d + 7$ mph. The time it takes for Doreen to travel 48 miles is $48/d$ hours, and the time it takes for Sue to travel 38 miles is $38/(d + 7)$ hours. According to the problem, Doreen takes 2 hours more than Sue, so we have the equation $48/d = 38/(d + 7) + 2$. Solving this equation for $d$ gives us Doreen's speed as 12 mph, which makes Sue's speed 19 mph. Plugging these speeds back into their respective distance/time equations gives us the times: Doreen travels for 4 hours (48 miles / 12 mph) and Sue travels for 2 hours (38 miles / 19 mph).

Question 10

Solve the equation. $\frac { x } { x - 4 } - \frac { 8 } { x + 12 } = \frac { 149 } { x ^ { 2 } + 8 x - 48 }$

Accepted Answer

The answer of Solve the equation. \(\frac { x }...

Question 11

Solve the equation for the variable x . $y = - \frac { v } { d } x + \frac { j } { m }$

Accepted Answer

To solve for x, we want to isolate it on one side of the equation. First, we can multiply both sides by -vd to get:
-yvd = vx - jd
Then, we can add jd to both sides to get:
jd - yvd = vx
Finally, we can divide both sides by vm to get:
x = (dj - dmy) / vm, which matches choice A.

Question 12

Solve each equation for the variable y . $- \frac { 5 } { x - 20 } = \frac { 9 } { y - 1 }$

Accepted Answer

First, cross-multiply to obtain $-5(y-1)=9(x-20)$. Expanding gives $-5y+5=9x-180$, so $y=\boxed{-9x/5+37}$.

Question 13

If two inlet pipes are both open, they can fill a pool in 3hours and 36 minutes. One of the pipes can fill the pool by itself in 6 hours. How long, in hours, would it take the other pipe to fill the pool by itself?

Accepted Answer

The total time taken by both pipes to fill the pool is 3 hours and 36 minutes, which is 3.6 hours. If one pipe fills the pool in 6 hours, we can use the formula for combined work rates: 1/A + 1/B = 1/T, where A and B are the times taken by each pipe alone, and T is the time taken together. Substituting the known values, we get 1/6 + 1/B = 1/3.6. Solving for B gives B = 9 hours.

Question 14

Solve each equation for the variable N . $\frac { N } { S } = \frac { T } { S + T }$

Accepted Answer

To solve for N, we can cross-multiply to get:
$$N(S+T) = T \cdot N$$
Then we can divide both sides by (S+T):
$$N = \frac{ T } { S + T } \cdot N $$
Finally, we can divide both sides by $\frac{T}{S+T}$ to get:
$$N = \frac{ S T } { S + T }$$
Therefore, the correct choice is C.

Question 15

Solve each equation for the variable Q . $I = \frac { 300 Q } { D }$

Accepted Answer

To solve for Q, multiply both sides by D and then divide by 300: $Q = \frac{DI}{300}$.

Question 16

Solve the equation. $4 + \frac { 12 } { t - 5 } = \frac { 12 } { t ( t - 5 ) }$

Accepted Answer

The answer of Solve the equation. \(4 + \frac {...

Question 17

Solve each equation for the variable x . $y = \frac { 2 } { 3 } x + \frac { 5 } { 18 }$

Accepted Answer

To solve for x, we need to isolate x on one side of the equation. 
Starting with the given equation:
$$y = \frac{2}{3}x + \frac{5}{18}$$
Subtracting $\frac{5}{18}$ from both sides:
$$y - \frac{5}{18} = \frac{2}{3}x$$
Multiplying both sides by $\frac{3}{2}$ (reciprocal of $\frac{2}{3}$):
$$\frac{3}{2}(y-\frac{5}{18}) = x$$
Simplifying:
$$x=\frac{3}{2}y-\frac{5}{12}$$
Therefore, the best choice is A.

Question 18

Solve the equation. $\frac { 5 y - 91 } { 10 y ^ { 2 } + 11 y - 6 } - \frac { 3 } { 2 y + 3 } = \frac { 5 } { 5 y - 2 }$

Accepted Answer

To solve the equation, first find a common denominator, which is $10y^2 + 11y - 6$. This denominator can be factored into $(2y + 3)(5y - 2)$. Then, solve the equation by combining like terms and simplifying. After simplification, you'll find that $y = -5$ is the solution that satisfies the original equation.

Question 19

Barry can do a certain job in 2 hours, whereas it takes Sanchez 3 hours to do the same job. How long would it take them to do the job working together?

Accepted Answer

Let's say the job requires painting a room. Barry can paint the room in 2 hours, which means he can paint 1/2 of the room per hour. Sanchez can paint the same room in 3 hours, which means he can paint 1/3 of the room per hour.
Working together, Barry and Sanchez can paint (1/2 + 1/3) = 5/6 of the room per hour. Therefore, they will complete the entire job in 6/5 hours or 1 hour and 12 minutes, which is 1 1/5 hours or 1.2 hours. The closest option is B.

Question 20

Felipe jogs for 12 miles and then walks another 12 miles. He jogs $2 \frac { 1 } { 2 }$ miles per hour faster than he walks, and the entire distance of 24 miles takes 11 hours. Find the rate at which he walks and the rate at which he jogs.

Accepted Answer

The answer of Felipe jogs for 12 miles and then...

Quiz 9: Rational Expressions

An inlet pipe can fill a tank in 15 minutes. A drain can empty the tank in 20 minutes. If the tank is empty and both the pipe and drain are open, how long will it take before the tank overflows? __________ minutes

Kent drives his Mazda 250 miles in the same time that Dave drives his Nissan 225 miles. If Kent averages 5 miles per hour faster than Dave, find their rates.

It takes Amy twice as long to deliver papers as it does Nancy. How long, in minutes, would it take each girl to deliver the papers by herself if they can deliver the papers together in 20 minutes?

Solve each equation for the variable y . y−bx=m\frac { y - b } { x } = mxy−b​=m

Solve the equation. xx−3−4x+7=52x2+4x−21\frac { x } { x - 3 } - \frac { 4 } { x + 7 } = \frac { 52 } { x ^ { 2 } + 4 x - 21 }x−3x​−x+74​=x2+4x−2152​

An inlet pipe can fill a tank in 10 minutes. A drain can empty the tank in 15 minutes. If the tank is empty and both the pipe and drain are open, how long will it take before the tank overflows?

If two inlet pipes are both open, they can fill a pool in 2 hours and 24 minutes. One of the pipes can fill the pool by itself in 6 hours. How long, in hours, would it take the other pipe to fill the pool by itself? __________ hours

Solve the equation. x5x2+6x−x5x2+11x+6=4x2+x\frac { x } { 5 x ^ { 2 } + 6 x } - \frac { x } { 5 x ^ { 2 } + 11 x + 6 } = \frac { 4 } { x ^ { 2 } + x }5x2+6xx​−5x2+11x+6x​=x2+x4​

To travel 38 miles, it takes Sue, riding a moped, 2 hours less time than it takes Doreen to travel 48 miles riding a bicycle. Sue travels 7 miles per hour faster than Doreen. Find the times and rates of both girls.

Solve the equation. xx−4−8x+12=149x2+8x−48\frac { x } { x - 4 } - \frac { 8 } { x + 12 } = \frac { 149 } { x ^ { 2 } + 8 x - 48 }x−4x​−x+128​=x2+8x−48149​

Solve the equation for the variable x . y=−vdx+jmy = - \frac { v } { d } x + \frac { j } { m }y=−dv​x+mj​

Solve each equation for the variable y . −5x−20=9y−1- \frac { 5 } { x - 20 } = \frac { 9 } { y - 1 }−x−205​=y−19​