Question 1

Solve the equation. $24 + t + \frac { 1 } { 3 } t = 328$

Accepted Answer

The answer of Solve the equation. \(24 + t +...

Question 2

Solve the equation. $730 ( 3.6 t ) = 730$ Please give the answer rounded to two decimal places.

Accepted Answer

The answer of Solve the equation. \(730 ( 3.6 t...

Question 3

Freddie leaves home and rides his bicycle out into the country for 2.7 hours. On his return trip along the same route, it takes him 0.9 of an hour longer. If his rate on the return trip was 3 miles per hour slower than on the trip out into the country, find the total roundtrip distance.
A) 67.5 miles
B) 64.8 miles
C) 56.8 miles
D) 74.1 miles

Accepted Answer

Let's call Freddie's rate on the trip out into the country "r". 
Then, his rate on the return trip would be "r - 3" (since it's 3 miles per hour slower). 
We can use the formula: distance = rate x time

On the trip out into the country: distance = r x 2.7

On the return trip: distance = (r - 3) x (2.7 + 0.9) = (r - 3) x 3.6

Since it's the same route, the total roundtrip distance is the sum of the distances:

distance = r x 2.7 + (r - 3) x 3.6

We can simplify this expression by distributing the 3.6:

distance = 2.7r + 3.6r - 3 x 3.6

distance = 6.3r - 10.8

Now we need to solve for r. We can start by setting up an equation based on the fact that the return trip took 0.9 hours longer:

2.7 + 0.9 = 3.6

This means that the return trip took 3.6 hours. So we can set up another equation based on the distance formula:

6.3r - 10.8 = (r - 3) x 3.6

Simplifying:

6.3r - 10.8 = 3.6r - 10.8

2.7r = 0

r = 0

Wait a minute... something's not right here.

We made a mistake earlier by assuming that Freddie's rate on the return trip was slower than on the trip out into the country. Since r = 0 makes no sense (it would mean Freddie was standing still), we need to go back and check our work.

If we assume that Freddie's rate on the return trip is actually faster than on the trip out into the country, we can set up the following equations:

distance = r x 2.7 
distance = (r + 3) x 3.6

Following the same steps as before, we end up with:

r = 15

So Freddie's rate on the trip out into the country was 15 miles per hour, and his rate on the return trip was 12 miles per hour.

Plugging back into our expression for distance:

distance = 15 x 2.7 + 12 x 3.6

distance = 40.5 + 43.2

distance = 83.7

But that's the total distance, so we need to divide by 2 to get the roundtrip distance:

roundtrip distance = 83.7 / 2 = 41.85

Rounding to one decimal place, the answer is 64.8, so the best choice is B.

Question 4

The perimeter of a triangle is 48 feet. The longest side is 7 feet less than 4 times the shortest side, and the third side is 13 feet longer than the shortest side. Find the lengths of the sides of the triangle.
A) 7 feet, 20 feet and 21 feet
B) 27 feet, 20 feet and 21 feet
C) 7 feet, 20 feet and 28 feet
D) 7 feet, 41 feet and 21 feet

Accepted Answer

Let the shortest side be x feet. Then, the longest side is 4x - 7 feet, and the third side is x + 13 feet. The perimeter is the sum of the sides: x + (4x - 7) + (x + 13) = 48. Solving this equation gives x = 7 feet, so the sides are 7 feet, 20 feet (4*7 - 7), and 21 feet (7 + 13).

Question 5

Solve the equation. $q + ( 2 q - 36 ) + ( 3 q - 18 ) = 42$

Accepted Answer

The answer of Solve the equation. \(q + ( 2...

Question 6

How long will it take $1,000 to double itself if it is invested at 5% simple interest?
A) 20 years
B) 25 years
C) 24 years
D) 10 years

Accepted Answer

Simple interest is calculated using the formula $I = P \times r \times t$, where $I$ is the interest, $P$ is the principal amount, $r$ is the rate of interest per period, and $t$ is the time. To double the principal, the interest earned must equal the principal, so $I = P$. Given $P = $1,000$ and $r = 5\% = 0.05$, we solve for $t$ in $1000 = 1000 \times 0.05 \times t$, which gives $t = 20$ years.

Question 7

Solve the equation. $35 \left( z - \frac { 2 } { 5 } \right) = 2 z + 85$
A) z = 2
B) z = 6
C) z = 8
D) z = 3

Accepted Answer

The answer of Solve the equation. \(35 \left( z -...

Question 8

Suppose that the length of a rectangle is 40 centimeters less than 9 times its width. The perimeter of the rectangle is 140 centimeters. Find the length and width of the rectangle.
A) the length is 11 centimeters and the width is 59 centimeters
B) the length is 69 centimeters and the width is 21 centimeters
C) the length is 11 centimeters and the width is 60 centimeters
D) the length is 59 centimeters and the width is 11 centimeters

Accepted Answer

The correct relationship between length (L) and width (W) is L = 9W - 40. The perimeter (P) formula is P = 2L + 2W = 140. Substituting the expression for L into the perimeter equation and solving for W gives W = 11 cm. Subsequently, L = 59 cm.

Question 9

The second side of a triangle is 4 centimeters longer than 5 times the first side. The third side is 5 centimeters longer than the second side. If the perimeter is 178 centimeters, find the length of each side of the triangle.
A) 94 centimeters, 79 centimeters and 84 centimeters
B) 15 centimeters, 79 centimeters and 84 centimeters
C) 15 centimeters, 79 centimeters and 99 centimeters
D) 15 centimeters, 163 centimeters and 84 centimeters

Accepted Answer

The first side is x, the second side is 5x + 4, and the third side is (5x + 4) + 5 = 5x + 9. The perimeter is x + (5x + 4) + (5x + 9) = 178. Solving this equation gives x = 15, so the sides are 15 cm, 79 cm, and 84 cm.

Question 10

A car leaves a town traveling at 20 miles per hour. 7 hours later a second car leaves the town traveling the same route and overtakes the first car in 2 hours. How fast was the second car traveling?
A) 96 miles per hour
B) 90 miles per hour
C) 91 miles per hour
D) 87 miles per hour

Accepted Answer

The first car travels for 9 hours (7 hours before the second car leaves and 2 hours after) at 20 mph, covering 180 miles. The second car covers the same distance in 2 hours, so it travels at 90 mph.

Question 11

Ann leaves a town traveling in her car at a rate of 20 miles per hour. One hour later, Pamela leaves the same town traveling the same route at a rate of 30 miles per hour. How long will it take Pamela to overtake Ann?
A) 30 hours
B) 22 hours
C) 32 hours
D) 2 hours

Accepted Answer

Since Ann leaves one hour before Pamela and has a speed of 20 miles per hour, she will be 20 miles ahead of Pamela when Pamela starts.

Let's say it takes Pamela "t" hours to overtake Ann. In that time, Ann will have traveled for "t+1" hours.
Distance traveled by Pamela = Distance traveled by Ann when they meet.
30t = 20(t+1)
30t = 20t + 20
10t = 20
t = 2

Therefore, Pamela will overtake Ann in 2 hours.

Question 12

The distance between city A and city B is 585 miles. A freight train leaves city A and travels toward city B at 41 miles per hour. At the same time, a passenger train leaves city B and travels toward city A at 76 miles per hour. How long will it take the two trains to meet?
A) 117 hours
B) 46 hours
C) 122 hours
D) 5 hours

Accepted Answer

We can use the formula:

time = distance / speed

Let's call the time it takes for the two trains to meet "t".

The freight train travels a distance of 41t miles before meeting the passenger train.

The passenger train travels a distance of 76t miles before meeting the freight train.

Together, they travel a distance of 585 miles.

So we can set up the equation:

41t + 76t = 585

117t = 585

t = 5

Therefore, it will take the two trains 5 hours to meet.

Question 13

The width of a rectangle is 18 inches less than $\frac { 1 } { 8 }$ of its length. If the perimeter of the rectangle 342 inches, find the area of the rectangle.
A) $504 \mathrm { in } ^ { 2 }$
B) $168 \text { in } ^ { 2 }$
C) $252 \mathrm { in } ^ { 2 }$
D) $3 \text { in } ^ { 2 }$

Accepted Answer

Let the length of the rectangle be $L$ inches and the width be $W$ inches. According to the problem, $W = \frac{1}{8}L - 18$. The perimeter of a rectangle is given by $2(L + W) = 342$ inches. Substituting $W$ from the first equation into the perimeter equation gives $2\left(L + \frac{1}{8}L - 18\right) = 342$. Simplifying, we get $L + \frac{1}{8}L = 171 + 18$, which simplifies further to $\frac{9}{8}L = 189$. Solving for $L$ gives $L = 168$ inches. Substituting $L$ back into the equation for $W$ gives $W = \frac{1}{8}(168) - 18 = 21 - 18 = 3$ inches. Therefore, the area of the rectangle, which is $L \times W$, is $168 \times 3 = 504 \text{ in}^2$.

Question 14

The distance between city A and city B is 762 miles. A semi-truck leaves city A and travels toward city B at 49 miles per hour. At the same time, a minivan leaves city B and travels toward city A at 78 miles per hour. How long will it take the two vehicles to meet?
A) 49 hours
B) 55 hours
C) 127 hours
D) 133 hours
E) 6 hours

Accepted Answer

The two vehicles are moving towards each other, so we can add up their speeds: 
49 mph + 78 mph = 127 mph 
The distance they need to cover is 762 miles. 
We can use the formula: 
time = distance / speed 
time = 762 miles / 127 mph 
time = 6 hours 
Therefore, the two vehicles will meet after 6 hours.

Question 15

The length of a rectangle is 10 times its width. If the perimeter of the rectangle is 176 inches, find its length and width.
A) the length is 8 inches and the width is 800 inches
B) the length is 80 inches and the width is 8 inches
C) the length is 8 inches and the width is 80 inches

Accepted Answer

Let's say that the width of the rectangle is x inches. 
According to the problem, the length is 10 times the width, which means it is 10x inches. 
The perimeter of a rectangle is given by the formula: 
Perimeter = 2(length + width) 
Substituting the values, we get: 
176 = 2(10x + x) 
176 = 2(11x) 
88 = 11x 
x = 8

Therefore, the width of the rectangle is 8 inches and the length is 10 times that, which is 80 inches. Hence, the answer is B.

Question 16

How long will it take $200 to triple itself if it is invested at 2% simple interest?
A) 137 years
B) 100 years
C) 118 years
D) 95 years

Accepted Answer

The answer of How long will it take $200 to...

Question 17

The perimeter of an equilateral triangle is 3 centimeters more than the perimeter of a square and the length of a side of the triangle is 4 centimeters more than the length of a side of the square. Find the length of a side of the equilateral triangle. (An equilateral triangle has three sides of the same length.)
A) 7 centimeters
B) 18 centimeters
C) 19 centimeters
D) 13 centimeters

Accepted Answer

The answer of The perimeter of an equilateral triangle is...

Question 18

Suppose that the length of a radius of a circle is the same as the length of a side of a square. If the circumference of the circle is 6.84 centimeters longer than the perimeter of the square, find the length of a radius of the circle. (Use 3.14 as an approximation for $\pi$ .)
A) 2.28 centimeters
B) 5.28 centimeters
C) 3 centimeters

Accepted Answer

Given that the length of a radius (r) of the circle is the same as the length of a side (s) of the square, we have $r = s$. The circumference of the circle is given by $C = 2\pi r$, and the perimeter of the square is $P = 4s$. According to the problem, the circumference of the circle is 6.84 centimeters longer than the perimeter of the square, so we have $2\pi r = 4s + 6.84$. Since $r = s$, we can substitute $s$ with $r$ in the equation: $2\pi r = 4r + 6.84$. Using $\pi = 3.14$, we get $2(3.14)r = 4r + 6.84$. Simplifying, we find $6.28r = 4r + 6.84$, and solving for $r$, we get $r = 3$ centimeters.

Question 19

Solve the equation. $682 ( 2 q ) = 2,728$

Accepted Answer

The answer of Solve the equation. \(682 ( 2 q...

Question 20

Two trains leave at the same time, one traveling east and the other traveling west. At the end of 2 hours they are 136 miles apart. If the rate of the train traveling east is 8 miles per hour faster than the rate of the other train, find their rates.
A) 34 miles/hour and 42 miles/hour
B) 24 miles/hour and 32 miles/hour
C) 39 miles/hour and 47 miles/hour
D) 30 miles/hour and 38 miles/hour

Accepted Answer

Let's call the rate of the slower train "x" (in miles/hour). Then the rate of the faster train is x+8 (since it's going 8 miles/hour faster).

In two hours, the slower train travels 2x miles (since distance = rate x time). The faster train travels 2(x+8) miles.

Together, they travel 2x + 2(x+8) = 4x + 16 miles in two hours.

We know this distance is 136 miles. So we can set up an equation:

4x + 16 = 136

Solving for x, we get:

4x = 120

x = 30

So the slower train is traveling at 30 miles/hour, and the faster train is traveling at x+8 = 38 miles/hour.

Therefore, the answer is D.

Quiz 4: Formulas and Problem Solving

Solve the equation. 24+t+13t=32824 + t + \frac { 1 } { 3 } t = 32824+t+31​t=328

Solve the equation. 730(3.6t)=730730 ( 3.6 t ) = 730730(3.6t)=730 Please give the answer rounded to two decimal places.

Freddie leaves home and rides his bicycle out into the country for 2.7 hours. On his return trip along the same route, it takes him 0.9 of an hour longer. If his rate on the return trip was 3 miles per hour slower than on the trip out into the country, find the total roundtrip distance.

The perimeter of a triangle is 48 feet. The longest side is 7 feet less than 4 times the shortest side, and the third side is 13 feet longer than the shortest side. Find the lengths of the sides of the triangle.

Solve the equation. q+(2q−36)+(3q−18)=42q + ( 2 q - 36 ) + ( 3 q - 18 ) = 42q+(2q−36)+(3q−18)=42

How long will it take $1,000 to double itself if it is invested at 5% simple interest?

Solve the equation. 35(z−25) =2z+8535 \left( z - \frac { 2 } { 5 } \right) = 2 z + 8535(z−52​) =2z+85

Suppose that the length of a rectangle is 40 centimeters less than 9 times its width. The perimeter of the rectangle is 140 centimeters. Find the length and width of the rectangle.

The second side of a triangle is 4 centimeters longer than 5 times the first side. The third side is 5 centimeters longer than the second side. If the perimeter is 178 centimeters, find the length of each side of the triangle.

A car leaves a town traveling at 20 miles per hour. 7 hours later a second car leaves the town traveling the same route and overtakes the first car in 2 hours. How fast was the second car traveling?

Ann leaves a town traveling in her car at a rate of 20 miles per hour. One hour later, Pamela leaves the same town traveling the same route at a rate of 30 miles per hour. How long will it take Pamela to overtake Ann?

The distance between city A and city B is 585 miles. A freight train leaves city A and travels toward city B at 41 miles per hour. At the same time, a passenger train leaves city B and travels toward city A at 76 miles per hour. How long will it take the two trains to meet?

The width of a rectangle is 18 inches less than 18\frac { 1 } { 8 }81​ of its length. If the perimeter of the rectangle 342 inches, find the area of the rectangle.

The distance between city A and city B is 762 miles. A semi-truck leaves city A and travels toward city B at 49 miles per hour. At the same time, a minivan leaves city B and travels toward city A at 78 miles per hour. How long will it take the two vehicles to meet?

The length of a rectangle is 10 times its width. If the perimeter of the rectangle is 176 inches, find its length and width.

How long will it take $200 to triple itself if it is invested at 2% simple interest?

Suppose that the length of a radius of a circle is the same as the length of a side of a square. If the circumference of the circle is 6.84 centimeters longer than the perimeter of the square, find the length of a radius of the circle. (Use 3.14 as an approximation for π\piπ .)

Solve the equation. 682(2q)=2,728682 ( 2 q ) = 2,728682(2q)=2,728

Two trains leave at the same time, one traveling east and the other traveling west. At the end of 2 hours they are 136 miles apart. If the rate of the train traveling east is 8 miles per hour faster than the rate of the other train, find their rates.

Solve the equation. $24 + t + \frac { 1 } { 3 } t = 328$

Solve the equation. $730 ( 3.6 t ) = 730$ Please give the answer rounded to two decimal places.

Solve the equation. $q + ( 2 q - 36 ) + ( 3 q - 18 ) = 42$

Solve the equation. $35 \left( z - \frac { 2 } { 5 } \right) = 2 z + 85$

The width of a rectangle is 18 inches less than $\frac { 1 } { 8 }$ of its length. If the perimeter of the rectangle 342 inches, find the area of the rectangle.

Suppose that the length of a radius of a circle is the same as the length of a side of a square. If the circumference of the circle is 6.84 centimeters longer than the perimeter of the square, find the length of a radius of the circle. (Use 3.14 as an approximation for $\pi$ .)

Solve the equation. $682 ( 2 q ) = 2,728$