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

Question

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.