Question

Deangelo has $7 worth of dimes and quarters in a jar. He has 7 more quarters than dimes.

How many of each coin does he have?

Answer 1

Let \( d \) represent the number of dimes. Since Deangelo has 7 more quarters than dimes, we can represent the number of quarters as \( q = d + 7 \).

The total value of the coins can be expressed in cents. Since dimes are worth 10 cents and quarters are worth 25 cents, we have:

\[
10d + 25q = 700
\]

Substituting \( q \) in terms of \( d \):

\[
10d + 25(d + 7) = 700
\]

Now, expanding and simplifying the equation:

\[
10d + 25d + 175 = 700
\]

Combining like terms:

\[
35d + 175 = 700
\]

Next, we subtract 175 from both sides:

\[
35d = 525
\]

Now, we divide by 35:

\[
d = 15
\]

Now, we can find the number of quarters by substituting \( d \) back into our expression for \( q \):

\[
q = d + 7 = 15 + 7 = 22
\]

Thus, Deangelo has \( 15 \) dimes and \( 22 \) quarters. To double-check the total value:

- The value of the dimes: \( 15 \times 10 = 150 \) cents.
- The value of the quarters: \( 22 \times 25 = 550 \) cents.

Adding these values gives:

\[
150 + 550 = 700 \text{ cents} = 7 \text{ dollars}
\]

This confirms our solution is correct. Therefore, the final answer is:

- Dimes: \( 15 \)
- Quarters: \( 22 \)

Answer 2

A rectangle has a length of 3.6 in. and a perimeter of 16.8 in.

Enter the width of the rectangle, in decimal form.

Answer 3

To find the width of the rectangle, we can use the formula for the perimeter of a rectangle, which is given by:

\[
P = 2L + 2W
\]

where \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width. We know the perimeter \( P = 16.8 \) inches and the length \( L = 3.6 \) inches.

Substituting the known values into the perimeter formula gives us:

\[
16.8 = 2(3.6) + 2W
\]

Calculating \( 2(3.6) \):

\[
2(3.6) = 7.2
\]

So the equation simplifies to:

\[
16.8 = 7.2 + 2W
\]

Next, we isolate \( 2W \) by subtracting \( 7.2 \) from both sides:

\[
16.8 - 7.2 = 2W
\]

Calculating the left side:

\[
9.6 = 2W
\]

Now, divide both sides by \( 2 \) to solve for \( W \):

\[
W = \frac{9.6}{2} = 4.8
\]

Therefore, the width of the rectangle is:

\[
\boxed{4.8} \text{ inches}
\]

Answer 4

Train A and train B leave a central station at the same time. They travel the same speed, but in opposite directions, with train A heading towards station A, and train B heading towards station B. Train A reaches station A after 3 h. Train B reaches station B after 2 1/4 h. Station A and Station B are 420 mi apart.

What is the rate of the trains?

Answer 5

To determine the rate of the trains, we first need to find the distances they each traveled until they reached their respective stations.

### Calculation of Distances

**Distance traveled by Train A:**
- Train A travels for 3 hours to reach Station A.

Based on the formula for distance:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
we can express the distance traveled by Train A as:
\[
\text{Distance A} = r \times 3 \quad (1)
\]

**Distance traveled by Train B:**
- Train B travels for \( 2 \frac{1}{4} \) hours, which can be converted to an improper fraction as \( 2.25 \) hours.

Thus, the distance traveled by Train B is:
\[
\text{Distance B} = r \times 2.25 \quad (2)
\]

### Relationship Between Distances

Since the total distance between Station A and Station B is 420 miles, we can express this as:
\[
\text{Distance A} + \text{Distance B} = 420
\]

Substituting equations (1) and (2) into the equation gives us:
\[
r \times 3 + r \times 2.25 = 420
\]

We can factor out \( r \):
\[
r (3 + 2.25) = 420
\]

Calculating \( 3 + 2.25 \):
\[
3 + 2.25 = 5.25
\]

Now the equation becomes:
\[
5.25r = 420
\]

### Solving for \( r \)

To find \( r \), divide both sides by \( 5.25 \):
\[
r = \frac{420}{5.25}
\]

Calculating \( \frac{420}{5.25} \):
\[
r = 80
\]

### Conclusion

The rate of the trains is \( \boxed{80} \) miles per hour.