Lesson 27 – Coin Problem

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.

Problem 60 – Relative Rate Problem

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

  1. One full circle = 360 degrees
  2. Displacement = rate • time
  3. 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

What are the units for the number 21.81? Minutes!

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


Problem

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.