Last winter Armand had StartFraction 5 Over 6 EndFraction of a row of stacked logs. At the end of the winter he had StartFraction 8 Over 15 EndFraction of the same row left. How much wood did he burn over the winter?
Armand burned StartFraction 5 Over 6 minus 8 over 15 EndFraction of the row of stacked logs. Simplifying the expression within the parentheses, we have:
StartFraction 5 Over 6 EndFraction minus StartFraction 8 Over 15 EndFraction
To subtract these fractions, we need a common denominator. The smallest number that both 6 and 15 divide into is 30, so we can rewrite the fractions with denominators of 30:
StartFraction 5 Over 6 EndFraction minus StartFraction 8 Over 15 EndFraction = StartFraction 25 Over 30 EndFraction minus StartFraction 16 Over 30 EndFraction
Now we can subtract the numerators:
StartFraction 25 Over 30 EndFraction minus StartFraction 16 Over 30 EndFraction = StartFraction 9 Over 30 EndFraction
Simplifying the fraction, we have:
StartFraction 9 Over 30 EndFraction = StartFraction 3 Over 10 EndFraction
So Armand burned 3/10 of the row of stacked logs over the winter.
At the beginning of the week, Naveen has 87 trading cards. On Monday, he gets 9 more. On Tuesday, he gives 16 away. On Wednesday, he is given 3 by a friend. On Thursday, he sells all of his cards to a collector. The collector pays him a rate of c for every card that Naveen has. If Naveen earns $124.50 from the sale, what is c, the rate that the collector pays for each card?
$1.08
$1.50
$41.50
$83.00
On Monday, Naveen has 87 + 9 = 96 trading cards.
On Tuesday, he has 96 – 16 = 80 trading cards.
On Wednesday, he has 80 + 3 = 83 trading cards.
On Thursday, he sells all 83 cards at a rate of c per card and earns $124.50.
Therefore, we can set up the equation:
83c = 124.50
Solving for c, we get:
c = 124.50 / 83 = 1.50
Therefore, the rate that the collector pays for each card is $1.50.
What is the first step in solving the equation 3.5 n + 6.4 = 42.5?
Add 6.4 to each side of the equation.
Subtract 6.4 from each side of the equation.
Multiply each side of the equation by 3.5.
Divide each side of the equation by 6.4.
Subtract 6.4 from each side of the equation.
What do the four yellow tiles in the model of the equation below represent?
3 long x tiles and 4 square 1 tiles = 2 negative 2 tiles
–4
4
–4x
4x
The four yellow tiles in the model represent 4 square 1 tiles.
Which expression is equivalent to 9 p minus 3 p + 2?
8p
14p
6 p + 2
9 p minus 1
The expression that is equivalent to 9 p minus 3 p + 2 is 6 p + 2.
Explanation:
9 p minus 3 p simplifies to 6 p. So we can rewrite the expression as:
6 p + 2
This is Daniela’s work to solve the equation 13 x + 42 = 276.
13 x + 42 = 276. 13 x = 318. x = 24 and StartFraction 6 Over 13 EndFraction.
Which property did Daniela need to use to avoid her error?
division property of equality
multiplication property of equality
subtraction property of equality
identity property of equality
division property of equality.
Solve the equation using the distributive property and properties of equality.
Negative 5 (a + 3) = negative 55
What is the value of a?
Negative 14
Negative 8
8
14
We can use the distributive property to simplify the left-hand side of the equation:
Negative 5 (a + 3) = negative 5a - 15
Now we can substitute this expression into the original equation and solve for a:
negative 5a - 15 = negative 55
Adding 15 to each side:
negative 5a = negative 40
Dividing each side by negative 5:
a = 8
Therefore, the value of a is 8.
The perimeter of a square is equal to the perimeter of an equilateral triangle. The length of a side of the square is given by x, and the length of a side of the equilateral triangle is given by x + 1. Which equation can be used to find the value of x?
x = 3 (x + 1)
3 x = 4 (x + 1)
4 x = 3 (x + 1)
x = x + 1
4 x = 3 (x + 1)
Zahra wants the equation below to have an infinite number of solutions when the missing number is placed in the box.
Box (x minus 3) + 2 x = negative (x minus 5) + 4
Which number should she place in the box?
–3
–1
1
3
To have an infinite number of solutions, the equation needs to simplify to an identity.
Expanding both sides of the equation, we have:
x - 3 + 2x = -x + 5 + 4
Simplifying the right-hand side:
x - 3 + 2x = -x + 9
Combining like terms on the left-hand side:
3x - 3 = -x + 9
Adding x and 3 to each side:
4x = 12
Dividing each side by 4, we get:
x = 3
Therefore, if Zahra places 3 in the box, the equation will have an infinite number of solutions.
Yuto solved the equation below.
Negative 2 (x + 5) = negative 2 (x minus 2) + 5
What is the solution to Yuto’s equation?
–10
9
no solution
infinitely many solutions
We can start by using the distributive property to simplify both sides of the equation:
-2(x + 5) = -2(x - 2) + 5
-2x - 10 = -2x + 4 + 5
Simplifying the right-hand side:
-2x - 10 = -2x + 9
Subtracting -2x from both sides:
-10 = 9
This is a contradiction, which means that the equation has no solution.
Therefore, there is no solution to Yuto's equation.
Consider the solution to the linear equation.
5 (x + 6) = 50. 5 x + 30 = 50. 5 x = 20. x = 4.
Which describes the inverse operations used after the distributive property?
addition then division
subtraction then division
multiplication then subtraction
division then addition
subtraction then division.
Which expression is equivalent to 7 b + 4 b minus 1 b?
2b
4b
10b
12b