Question 1

The total cost of three automobile batteries is $182. The most expensive one is five times the cost of the least expensive. The third is $7 more than the least expensive. Find the cost of each battery.

Accepted Answer

Let the cost of the least expensive battery be x. Then, the most expensive battery would be 5x and the third would be x + 7. Thus, we have the equation x + 5x + x + 7 = 182 which simplifies to 7x = 175. Solving for x, we get x = 25. Therefore, the least expensive battery costs $25, the most expensive battery costs $125 (5x) and the third battery costs $32 (x + 7). Thus, the correct answer is A.

Question 2

James invests $4,700 in two savings accounts. One earns interest at 7% per year, while the other earns 12% per year. The total interest earned from both accounts after one year is $409. How much was originally deposited in each account?

Accepted Answer

Let x be the amount at 7% and y be the amount at 12%. The equations are x + y = 4700 and 0.07x + 0.12y = 409. Solving these equations gives x = 3100 and y = 1600.

Question 3

Cut a board 36 ft long into three pieces so that the longest piece will be twice as long as each of the other two of equal lengths. Find the length of each piece.

Accepted Answer

Let the length of the two equal pieces be x. According to the problem, the longest piece is twice as long as each of the other two pieces. Therefore, the third piece must be 2x. The sum of the three pieces must equal 36 feet, so we can write an equation: x + x + 2x = 36. Solving for x gives x = 9. Therefore, the length of the two equal pieces is 9 feet and the length of the third piece is 18 feet, which matches choice C.

Question 4

Solve the equation and check. $2 ( y + 6 ) = 6 ( y - 8 ) + 5$

Accepted Answer

The answer of Solve the equation and check. \(2 (...

Question 5

A rectangle is   as long as it is wide. Its perimeter (the sum of the lengths of its sides) is 88 cm. Find its length and width.

Accepted Answer

The perimeter of a rectangle is given by the formula 2(length + width). Since the rectangle is as long as it is wide, both the length and width are the same, let's call them x. So, the perimeter is 2(x + x) = 4x. Given that the perimeter is 88 cm, we solve for x: 4x = 88, which gives x = 22. Therefore, both the length and width are 22 cm.

Question 6

Solve the equation and check. $4 ( 2 y - 5 ) + 3 ( 5 y + 35 ) = 25 ( 3 y + 5 ) - 162 y$

Accepted Answer

The answer of Solve the equation and check. \(4 (...

Question 7

Solve the equation and check. $11 x - 6 ( x + 8 ) + 51 = 0$

Accepted Answer

The answer of Solve the equation and check. \(11 x...

Question 8

The perimeter of a triangle is 111 ft. The lengths of two sides are the same. The length of the third side is 6 ft shorter than either of the other two sides. Find the lengths of the sides.

Accepted Answer

Let the equal sides be x. Then the third side is x - 6. The equation is 2x + (x - 6) = 111. Solving gives x = 39, so the equal sides are 39 ft and the third side is 33 ft.

Question 9

Solve the equation and check. $0.095 x - 0.285 + 0.001 ( 5 - x ) = 0.262$

Accepted Answer

The answer of Solve the equation and check. \(0.095 x...

Question 10

One side of a rectangular yard is bounded by the side of a house. The other three sides are to be fenced with 81 ft of fencing. The length of fence opposite the house is 15 ft less than either of the other two sides. Find its length and width.

Accepted Answer

The answer of One side of a rectangular yard is...

Question 11

Cut a 13-ft beam into two pieces so that one piece is 12 in. longer than the other. Find the length of the two pieces.

Accepted Answer

Let x be the length of the shorter piece, then the length of the longer piece is x + 12. We know that the sum of the two pieces is 13 ft, which is equivalent to 156 in. Therefore, we can write an equation: x + (x + 12) = 156. Simplifying this equation, we get 2x + 12 = 156, which leads to 2x = 144 and x = 72. So the length of the shorter piece is 72 in and the length of the longer piece is 84 in, which matches with option C.

Question 12

The length of a rectangle is 3 cm less than five times its width. Its perimeter is 102 cm. Find its length and width.

Accepted Answer

The perimeter of a rectangle is given by the formula $P = 2l + 2w$, where $l$ is the length and $w$ is the width. Given that the length $l = 5w - 3$ and the perimeter $P = 102$, we can substitute the expression for $l$ into the perimeter formula to get $102 = 2(5w - 3) + 2w$. Simplifying, we get $102 = 10w - 6 + 2w$, which simplifies further to $108 = 12w$, leading to $w = 9$. Substituting $w = 9$ back into the expression for $l$, we get $l = 5(9) - 3 = 45 - 3 = 42$. Therefore, the width is 9 cm and the length is 42 cm.

Question 13

A set of four built-in bookshelves in to be constructed in a room. The floor-to-ceiling clearance is 6 ft 1 in. Each shelf is 7 in. thick. An equal space is to be left between shelves. What space should there be between each shelf and the next? ( There is no shelf against the ceiling and no shelf on the floor. ) Please do not use the rules for calculating with measurements.

Accepted Answer

The answer of A set of four built-in bookshelves in...

Question 14

Distribute $399 among Peter, Betsy and Emma so that Betsy receives five times as much as Peter, and Emma receives three times as much as Betsy.

Accepted Answer

Let the amount Peter receives be $x. 
Then, Betsy receives $5x (five times as much as Peter) and Emma receives $15x (three times as much as Betsy). 
We can set up the following equation: 
x + 5x + 15x = 399 
21x = 399 
x = 19 
Therefore, Peter receives $19, Betsy receives $95 (five times as much as Peter), and Emma receives $285 (three times as much as Betsy). 
Thus, the correct choice is D.

Question 15

Solve the equation and check. $3 x - 7 + 11 ( 1 - x ) = - 6$

Accepted Answer

The answer of Solve the equation and check. \(3 x...

Question 16

Peter and Mary earn a total of $502 by working a total of 37 hours. If Peter earns $16/h and Mary earns $10/h, how many hours did each work?

Accepted Answer

Let's use algebra to solve the problem. Let x be the number of hours Peter worked and y be the number of hours Mary worked. Then we have two equations:

x + y = 37 (total number of hours worked)
16x + 10y = 502 (total amount earned)

We can simplify the first equation to get:

y = 37 - x

Substitute this into the second equation to get:

16x + 10(37 - x) = 502
Simplify and solve for x:
6x + 370 = 502
6x = 132
x = 22

Substitute x = 22 back into y = 37 - x to get:

y = 15

Therefore, Peter worked 22 hours and Mary worked 15 hours. The answer is D.

Question 17

Separate an order of 23 light fixture so that the number of fluorescent light fixture will be 19 fewer than twice the number of incandescent light fixtures.

Accepted Answer

Let x be the number of incandescent fixtures and y be the number of fluorescent fixtures. We have two equations: x + y = 23 and y = 2x - 19. Solving these gives x = 14 and y = 9.

Question 18

Solve the equation and check. $14 + 14 ( 3 - 2 x ) + 11 = 3 - x + 5 ( 2 - 3 x )$

Accepted Answer

The answer of Solve the equation and check. \(14 +...

Question 19

Saw a board 4 ft 1 in. long into six equal pieces. If the loss per cut is 1/5 in., how long will each piece be? Please do not use the rules for calculating with measurements.

Accepted Answer

The answer of Saw a board 4 ft 1 in....

Question 20

Distribute $270 into three parts so that the first part will be $\text { twice }$ as large as the second, and the third part will be $\text { four times }$ as large as the sum of the other two.

Accepted Answer

Let the second part be $x$. Then, the first part is $2x$ and the third part is $4(2x + x) = 12x$. The total is $x + 2x + 12x = 15x = 270$, so $x = 18$. Therefore, the parts are $36, $18, and $216.

Quiz 6: Equations and Formulas

The total cost of three automobile batteries is $182. The most expensive one is five times the cost of the least expensive. The third is $7 more than the least expensive. Find the cost of each battery.

James invests $4,700 in two savings accounts. One earns interest at 7% per year, while the other earns 12% per year. The total interest earned from both accounts after one year is $409. How much was originally deposited in each account?

Cut a board 36 ft long into three pieces so that the longest piece will be twice as long as each of the other two of equal lengths. Find the length of each piece.

Solve the equation and check. 2(y+6)=6(y−8)+52 ( y + 6 ) = 6 ( y - 8 ) + 52(y+6)=6(y−8)+5

A rectangle is as long as it is wide. Its perimeter (the sum of the lengths of its sides) is 88 cm. Find its length and width.

Solve the equation and check. 4(2y−5)+3(5y+35)=25(3y+5)−162y4 ( 2 y - 5 ) + 3 ( 5 y + 35 ) = 25 ( 3 y + 5 ) - 162 y4(2y−5)+3(5y+35)=25(3y+5)−162y

Solve the equation and check. 11x−6(x+8)+51=011 x - 6 ( x + 8 ) + 51 = 011x−6(x+8)+51=0

The perimeter of a triangle is 111 ft. The lengths of two sides are the same. The length of the third side is 6 ft shorter than either of the other two sides. Find the lengths of the sides.

Solve the equation and check. 0.095x−0.285+0.001(5−x)=0.2620.095 x - 0.285 + 0.001 ( 5 - x ) = 0.2620.095x−0.285+0.001(5−x)=0.262