Interesting math problems

A compilation of interesting math problems used in real life. This is the math that you need to see in order to see the usefulness of math and as a result appreciate math.

Investment. Problem #1: An investor with $10000 invests part of the money at 8% and the rest at 12%. How much should be invested at each rate to yield 11% of the total amount? 
background knowledge: Percent and solving equations with fractions

Solution:

Let x be the part that is invested at 8%. The other part invested at 12% is 10000 - x.

x × 8% + (10000 - x) × 12% = 10000 × 11%

x ×
8 / 100
+
(10000 - x) ×
12 / 100
=
10000 ×
11 / 100


8x / 100
+
120000 - 12x / 100
=
110000 / 100


Multiply both sides by 100

100 ×
8x / 100
+
100 ×
120000 - 12x / 100
=
100 ×
110000 / 100


We get 8x + 120000 - 12x = 110000

12x - 8x = 120000 - 110000

4x = 10000

x = 2500 since 4 × 2500 = 10000

x ×
8 / 100
+
(10000 - x) ×
12 / 100
=
10000 ×
11 / 100


8x / 100
+
120000 - 12x / 100
=
110000 / 100


Multiply both sides by 100

100 ×
8x / 100
+
100 ×
120000 - 12x / 100
=
100 ×
110000 / 100


We get 8x + 120000 - 12x = 110000

12x - 8x = 120000 - 110000

4x = 10000

x = 2500 since 4 × 2500 = 10000

2500 should be invested at 8% and 7500 should be invested at 12%

Consumer Price Index. Problem #2:  Suppose  in 1960, the consumer price index was 70 while in 2015 it is 280. What monthly salary in 2015 would have the same purchasing power as a monthly salary of 800 dollars in 1960? 
background knowledge: Ratio and proportion

Solution:

Let x be the monthly salary in 2015. 

x / 800
  =
280 / 70


x × 70 = 280 × 800

70x = 224000

70 / 70
  x
=
224000 / 70


x = 3200

x / 800
  =
280 / 70


x × 70 = 280 × 800

70x = 224000

70 / 70
  x
=
224000 / 70


x = 3200

Business. Problem #3:  A bookstore made 11000 dollars selling 3000 math books and 1000 physics books. A customer bought 1 math book and 1 physics book and paid 7 dollars. What is the price of each book?
background knowledge: System of linear equations

Solution:

Let x be the price of the math book and y be the price of the physics book

x + y = 7

3000 × x + 1000 × y = 11000

Use x + y = 7 to solve for y

x - x + y = 7 - x

y = 7 - x

Substitute 7 - x for y in 3000 × x + 1000 × y = 11000

3000 × x + 1000 × (7 - x) = 11000

3000x + 7000 - 1000x = 11000

2000x = 11000 - 7000

2000x = 4000

x = 2 since 2000 × 2 = 4000

Since 2 + 5 = 7, y = 5

The price of a math book is 2 dollars and the price of a physics book is 5 dollars.


Ecology. Problem #4:
The pressure someone experiences as he or she dives deeper and deeper in the ocean increases linearly. On the surface, the pressure is close to 15 pounds per square inch. 33 feet below the surface, the pressure is 30 pounds. If 25000 pounds per sq inch can crush your bones, what depth is extremely dangerous for humans?

background knowledge: Linear equation

Solution:

First, model the pressure ( p ) in terms of depth ( d ) with a linear equation.

We will find the equation p = md + b

Use (0, 15) and (33, 30) to find m

m =
30 - 15 / 33 - 0

m =
15 / 33
= 0.45


p = 0.45d + b

Use (0, 15) to find b

15 = 0.45 × 0 + b

15 = b

p = 0.45d + 15

25000 = 0.45d + 15

25000 - 15 = 0.45d + 15 - 15

24985 = 0.45d

d =
24985 / 0.45
= 55522 feet

Keep in mind though that about 1000 feet beneath the ocean, it is already very dangerous. 

Demography. Problem #5: The population in the state of Florida right now is 19.9 million. The latest format for passenger plates in the state of florida is AAA A00 through ZZZ Z99. If the letter o cannot be used when making plates, will the state have enough license plate numbers for its residents?
background knowledge: Multiplicative principle ( Counting )

Solution:

The format shows that the first 4 characters are letters and the last 2 characters are numbers. There are 26 characters in the Enlgish alphabet. Since the letter o cannot be used, the choice is now 25. There are 10 characters for the numbers 0-9.

Notice also that numbers or letters can be repeated!

For the first letter, there are 25 choices

For the second letter, there are 25 choices
For the third letter, there are 25 choices
For the fourth letter, there are 25 choices
For the first number, there are 10 choices
For the second number, there are 10 choices

The counting principle says to multiply all the choices

Numbers of license plates = 25 × 25 × 25 × 25 × 10 × 10 = 39,062,500

Therefore, there will be more than enough license plates.

The interesting math problems above were put here to show you how math is applied in real life. It is my hope that you will love math as you explore them. I will update this page with more and more interesting math problems. Keep coming back!

                   More interesting math problems

Supply and demand.  Problem #6: Read problem
Background knowledge: Inverse variation, slope-intercept form, solve quadratic equations.





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