12(1.2x+4.65)=15(x+3.5)

Answer 1

To solve the equation \( 12(1.2x + 4.65) = 15(x + 3.5) \), we start by distributing both sides.

Step 1: Distribute on both sides.

Left side: \[ 12(1.2x) + 12(4.65) = 14.4x + 55.8 \]

Right side: \[ 15(x) + 15(3.5) = 15x + 52.5 \]

Now the equation looks like: \[ 14.4x + 55.8 = 15x + 52.5 \]

Step 2: Move all terms involving \( x \) to one side and constant terms to the other side.

Subtract \( 14.4x \) from both sides: \[ 55.8 = 15x - 14.4x + 52.5 \]

This simplifies to: \[ 55.8 = 0.6x + 52.5 \]

Subtract \( 52.5 \) from both sides: \[ 55.8 - 52.5 = 0.6x \]

This simplifies to: \[ 3.3 = 0.6x \]

Step 3: Solve for \( x \).

To isolate \( x \), divide both sides by \( 0.6 \): \[ x = \frac{3.3}{0.6} \]

Calculating this gives: \[ x = 5.5 \]

So the solution to the equation \( 12(1.2x + 4.65) = 15(x + 3.5) \) is: \[ \boxed{5.5} \]

Answer 2

A group of 6 friends went ice-skating. They each paid an admission fee of $x and $2.50 for hot chocolate. Another group of 5 friends went ice-skating at a different rink, and they paid 1.5 times as much for admission as the first group. The group of 5 friends spent $2.25 each for hot chocolate. The two groups paid the same total amount. What did each of the first group of friends pay for admission?

Answer 3

Let's denote the admission fee that each of the first group of friends paid as \( x \). Since there are 6 friends in that group, the total amount they paid for admission is:

\[ 6x \]

Each friend in the first group also paid $2.50 for hot chocolate, so the total amount spent by the first group on hot chocolate is:

\[ 6 \times 2.50 = 15.00 \]

Thus, the total amount spent by the first group on both admission and hot chocolate is:

\[ 6x + 15.00 \]

For the second group of 5 friends, they paid 1.5 times as much for admission as the first group. Therefore, the admission fee per person for the second group is:

\[ 1.5x \]

The total amount the second group paid for admission is:

\[ 5 \times 1.5x = 7.5x \]

Each friend in the second group paid $2.25 for hot chocolate, so the total amount spent by the second group on hot chocolate is:

\[ 5 \times 2.25 = 11.25 \]

So, the total amount spent by the second group on both admission and hot chocolate is:

\[ 7.5x + 11.25 \]

Since the two groups paid the same total amount, we can set the two total amounts equal to each other:

\[ 6x + 15.00 = 7.5x + 11.25 \]

Step 1: Rearranging the equation

To solve for \( x \), we can rearrange the equation by subtracting \( 6x \) and \( 11.25 \) from both sides:

\[ 15.00 - 11.25 = 7.5x - 6x \]

This simplifies to:

\[ 3.75 = 1.5x \]

Step 2: Solving for \( x \)

Next, we can solve for \( x \) by dividing both sides by \( 1.5 \):

\[ x = \frac{3.75}{1.5} \]

Calculating this gives:

\[ x = 2.5 \]

Thus, the admission fee that each of the first group of friends paid is:

\[ \boxed{2.50} \]

Answer 4

Emma walks around a rectangular city block. She walks a total of 380 yards. The length of the block is 10 yards longer than the width. What is the width of the city block?

Answer 5

Let's denote the width of the rectangular city block as \( w \) yards. According to the problem, the length of the block is \( w + 10 \) yards.

The perimeter \( P \) of a rectangle is given by the formula:

\[ P = 2(\text{length} + \text{width}) \]

Substituting for the length and width, we have:

\[ P = 2((w + 10) + w) \]

Simplifying this expression:

\[ P = 2(2w + 10) = 4w + 20 \]

We know from the problem that Emma walks a total of 380 yards around the block, thus:

\[ 4w + 20 = 380 \]

Step 1: Solve for \( w \)

First, we subtract 20 from both sides:

\[ 4w = 380 - 20 \] \[ 4w = 360 \]

Now, divide both sides by 4:

\[ w = \frac{360}{4} \] \[ w = 90 \]

Thus, the width of the city block is:

\[ \boxed{90 \text{ yards}} \]

Step 2: Check the dimensions

To ensure our calculations are correct, let's find the length:

\[ \text{Length} = w + 10 = 90 + 10 = 100 \text{ yards} \]

Now, we can calculate the perimeter again to verify:

\[ P = 2(\text{length} + \text{width}) = 2(100 + 90) = 2 \times 190 = 380 \text{ yards} \]

The calculated perimeter matches the total distance Emma walked, confirming our solution is correct. Thus, the width of the city block remains:

\[ \boxed{90 \text{ yards}} \]

Answer 6

2(4x-7)=66

Answer 7

To solve the equation \( 2(4x - 7) = 66 \), we can follow these steps:

Step 1: Distribute the left side

Distributing the \( 2 \) gives us:

\[ 8x - 14 = 66 \]

Step 2: Add 14 to both sides

Next, we add \( 14 \) to both sides of the equation:

\[ 8x - 14 + 14 = 66 + 14 \] \[ 8x = 80 \]

Step 3: Divide both sides by 8

Now, we divide both sides by \( 8 \) to solve for \( x \):

\[ x = \frac{80}{8} \] \[ x = 10 \]

Thus, the solution to the equation \( 2(4x - 7) = 66 \) is:

\[ \boxed{10} \]

Answer 8

-6(x+7)=18

Answer 9

To solve the equation \(-6(x + 7) = 18\), follow these steps:

Step 1: Distribute the \(-6\)

Distributing \(-6\) gives:

\[ -6x - 42 = 18 \]

Step 2: Add \(42\) to both sides

Next, add \(42\) to both sides of the equation:

\[ -6x - 42 + 42 = 18 + 42 \] \[ -6x = 60 \]

Step 3: Divide both sides by \(-6\)

Now, divide both sides by \(-6\) to solve for \(x\):

\[ x = \frac{60}{-6} \] \[ x = -10 \]

Thus, the solution to the equation \(-6(x + 7) = 18\) is:

\[ \boxed{-10} \]

Answer 10

5n-6(n+3)=14

Answer 11

To solve the equation \( 5n - 6(n + 3) = 14 \), follow these steps:

Step 1: Distribute \(-6\)

Distributing \(-6\) across \((n + 3)\) gives:

\[ 5n - 6n - 18 = 14 \]

Step 2: Combine like terms

Now, combine the \(n\) terms:

\[ (5n - 6n) - 18 = 14 \] \[ -n - 18 = 14 \]

Step 3: Add \(18\) to both sides

Add \(18\) to both sides to isolate \(-n\):

\[ -n = 14 + 18 \] \[ -n = 32 \]

Step 4: Multiply both sides by -1

To solve for \(n\), multiply both sides by \(-1\):

\[ n = -32 \]

Thus, the solution to the equation \( 5n - 6(n + 3) = 14 \) is:

\[ \boxed{-32} \]