Sammy has 13 coins of either quarters or dimes. The value of the coins totals $2.95. How many quarters and how many dimes does Sammy have?

*Introduce variables*

q = number of quarters

d = number of dimes

number of coins

13 = q + d or q = 13 – d

value of coins

295 = q • 25 + d • 10

**The two equations for algebra: **

13 = q + d

295 = q • 25 + d • 10

*Substitute for q *

(13 – d) • 25 + d • 10 = 295

*Distribute and Simplify*

325 – 25d + 10d = 295 → 325 – 15d = 295

d = 2

*Substitute 2 for d*

13 = 2 + q

q = 11

**Answer** – There are 11 quarters and 2 dimes.

Between 4 and 5 o’clock, at what time is the minute hand directly over the hour hand?

s = position of short hand in degrees

l = position of long hand in degrees

To begin the problem, we have to use 3 givens

- One full circle = 360 degrees
- Displacement = rate • time
- Speed =
__change in position in degrees__

time

Since there are 12 increments in the clock, each hour represents a span of 30 degrees.

*The long hand moves an entire circle in one hour*

speed of long hand = __ 360 deg__ = 6 degrees per minute

60 min

Beginning at 4 o’clock, the position is like this:

*******picture isn't included. Would be nice if pictures could be included*******

Final position = initial position + (rate • time)

long hand final = initial position + 6 degrees • t

l = 0 + 6 degrees • t

short hand final = initial position + 0.5 • t

s = 120 + 0.5 • t

When the short hand and the long hand are on top of each other, the position is the same for both. This means that s = l. You can substitute l for s.

l = 0 + 6 • t l = 120 + 0.5 • t

*Using linear combination, you multiply the bottom equation by negative 1*

l = 0 + 6 • t

__+ -l = -120 + -6 • t__

0 = -120 + 5.5t

*Simplify*

120 = 5.5t

t = 21.81

**Answer:** At 4:22, the handles are directly on top of each other.

Between 8 and 9 o’clock, when is the minute hand directly above the hour hand? All we need to do is change the starting point of the short hand.

l = 0 + 6 degrees • t

s = 240 + 0.5 degrees • t

*Again, replace l with s because the final position is the same*

s = 6t

s = 240 + 0.5 • t

Substitute 6t for s

6t = 240 + 0.5t

5.5t = 240

t = 43.63

Units are in minutes.

**Answer:** At 8:43, the hands are on top of one another.

**Bonus Question:** Separate the clock face into three equal parts so that the sum of the numbers within each part equals the same number.

- CyberProf(TM)