A. One board is one-third the length of another. Six times the sum of the length of the short board and -10 is equal to the length of the longer board decreased by 11 inches. Find the length of the longer board.
B. The length of a rectangle is 4 feet more than twice the width. Five times the width is the same as twice the length increased by 10 feet. Find the dimensions.
C. In February, Paul's electric bill was three dollars more than one-half his gas bill. if the electric bill was ninety-two dollars, what was the gas bill?
B. L=4+2x
W= x+5
5x=2(4+2x)+10
Solve for x and you should get 18.
Length: 40 ft
Width: 23 ft
C. Electric Bill= 3+1/2x
Gas Bill= x
92=3+1/2x
Answer: x=$178
A.
let the longer board be x
shorter board = x/3 or (1/3)x
6(x/3 -10) = x-11
2x - 60 = x -11
x = 49
long board = 49 ft
short board = 49/3 ft
check:
short board is 1/3 of the long one. check!
6(short board - 10)
= 6(49/3 - 10
= 98-60 = 38
longer decreased by 11
= 49-11 = 38
check!
A. Let's assume the length of the longer board is "x" inches. According to the problem, the length of the shorter board is one-third of the longer board, so it would be (1/3)x inches.
The problem states that six times the sum of the length of the short board (1/3)x inches and -10 is equal to the length of the longer board (x inches) decreased by 11 inches. So we can write the equation:
6((1/3)x - 10) = x - 11
Let's solve this equation step-by-step:
1. Distribute 6 to the terms inside the parentheses:
2x - 60 = x - 11
2. Move all terms containing "x" to one side by subtracting "x" from both sides:
2x - x = -11 + 60
x = 49
So the length of the longer board is 49 inches.
B. Let's assume the width of the rectangle is "w" feet. According to the problem, the length of the rectangle is 4 feet more than twice the width, so it would be 2w + 4 feet.
The problem states that five times the width is equal to twice the length increased by 10 feet. So we can write the equation:
5w = 2(2w + 4) + 10
Let's solve this equation step-by-step:
1. Distribute 2 to the terms inside the parentheses:
5w = 4w + 8 + 10
2. Combine like terms:
5w = 4w + 18
3. Move all terms containing "w" to one side by subtracting 4w from both sides:
5w - 4w = 18
w = 18
So the width of the rectangle is 18 feet.
Now, we can find the length by substituting the width value into the expression for the length:
Length = 2w + 4 = 2(18) + 4 = 36 + 4 = 40 feet
Therefore, the dimensions of the rectangle are 18 feet (width) and 40 feet (length).
C. Let's assume the gas bill is "g" dollars. According to the problem, the electric bill is three dollars more than one-half the gas bill, so it would be (1/2)g + 3 dollars.
The problem states that the electric bill was ninety-two dollars. So we can write the equation:
(1/2)g + 3 = 92
Let's solve this equation step-by-step:
1. Subtract 3 from both sides to isolate the term with "g":
(1/2)g = 92 - 3
(1/2)g = 89
2. Multiply both sides of the equation by 2 to get rid of the fraction:
2 * (1/2)g = 2 * 89
g = 178
Therefore, the gas bill was 178 dollars.
A. Let's assign variables to represent the lengths of the boards. Let's call the length of the short board x, and the length of the longer board y.
We are given that "One board is one-third the length of another," which means that x = (1/3)*y.
We're also given the equation: "Six times the sum of the length of the short board and -10 is equal to the length of the longer board decreased by 11 inches." Translating this into an equation, we have: 6*(x + (-10)) = y - 11.
Now, we can substitute x = (1/3)*y into the equation and solve for y:
6*(((1/3)*y) + (-10)) = y - 11
2y - 60 = y - 11
y = 49
Therefore, the length of the longer board is 49 inches.
B. Let's assign variables to represent the dimensions of the rectangle. Let's call the width x and the length y.
We're given that "The length of a rectangle is 4 feet more than twice the width." This can be expressed as y = 2x + 4.
We're also given the equation: "Five times the width is the same as twice the length increased by 10 feet." Translating this into an equation, we have: 5x = 2y + 10.
Now, we can substitute y = 2x + 4 into the equation and solve for x:
5x = 2*(2x + 4) + 10
5x = 4x + 8 + 10
5x = 4x + 18
x = 18
To find the length (y), we substitute the value of x back into the first equation:
y = 2x + 4
y = 2*18 + 4
y = 40
Therefore, the dimensions of the rectangle are 18 feet by 40 feet.
C. Let's assign variables to represent the electric bill and the gas bill. Let's call the electric bill x and the gas bill y.
We're given that "In February, Paul's electric bill was three dollars more than one-half his gas bill." This can be expressed as x = (1/2)*y + 3.
We're also given that "the electric bill was ninety-two dollars." This tells us that x = 92.
Now, we can substitute x = 92 into the first equation and solve for y:
92 = (1/2)*y + 3
(1/2)*y = 92 - 3
(1/2)*y = 89
y = 89 * 2
y = 178
Therefore, the gas bill was $178.