The most practical math for people to understand is undoubtedly high school math, rather than college math.

After all, how much calculus is used when you go grocery shopping, get your plumbing fixed, or you go skiing on the weekend?

## High School Math

You’ve got to love it. Here’s the first high school math problem.

**Problem 1:** Simplify the mathematical expression:

(x^{-2}y^{3})^{4} (x^{-3}y^{4})^{-2}

Simplifying the first parenthetical expression, we get

(x^{-8}y^{12})

It is the powers we multiply when powers are raised to powers.

Doing similarly with the second parenthetical expression, we get for that

(x^{6}y^{-8})

The equation now reads,

(x^{-8}y^{12}) (x^{6}y^{-8})

When we multiply numbers, we add and subtract powers. This gives,

(x^{-2}y^{4}) [Answer]

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**Problem 2:** 2/10 divided by n equals 3-1/2. What does n equal?

2/10 divided by n can be written 2/10/n. A fraction of this sort needs to be simplified. To do that, the n must come to the top next to the 2. Thus,

2/10 / n = 2n/10

So,

2n/10 = 3-1/2

Multiplying both sides by 10 and dividing both sides by 2 (which is the same thing as multiplying both sides by five), we get,

n = 17-1/2 [Answer]

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**Problem 3:** What is the answer to the problem: five over eight multiplied by four over nine?

The 5 and the 4 are called numerators. The 8 and the 9 are denominators. Numerators are divided by denominators.

In multiplication of fractions such as these, one multiplies the numerators. The same thing is done with the denominators. Thus, for 5/8 x 4/9,

5 x 4 = 20 and 8 x 9 = 72

The result is 20/72

This number can be simplified. First, write it out as a product of the smallest numbers possible for the numerator and the denominator…

20 = 2 x 2 x 5

72 = 2 x 2 x 2 x 9

For the top and the bottom, we can remove 2 x 2. So,

5/(2 x 9) = 5/18 [Answer]

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**Problem 4:** I am less than 3/4 but more than 1/2. My numerator and denominator are both square numbers. What fraction am I?

Let’s start with 16ths. 3/4 is 12/16. 1/2 is 8/16. Consider 9, 10, and 11/16. Of these, the number 9 is a square.

9/16 [Answer]

It is greater than 1/2 (which is the same as 8/16) and less than 3/4 (which is 12/16). 9 is 3 squared (3 x 3). 16 is 4 squared (4 x 4).

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**Problem 5:** How do you work this type problem:

f(x) = (1/x) – 2?

First, I convert the f(x) to y.

Simply:

y = (1/x) – 2

It is easy to draw a graph when in this form. For instance, when I plug in these numbers for x, I get out these other numbers for y:

(x, y)

(-1, -3)

(0, infinity)

(1, -1)

(2, -3/2)

If I wish the equation to look as simple as possible, I get rid of denominators.

y = (1/x) – 2 becomes,

x(y + 2) = 1 [Answer]

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**Problem 6:** Evaluate 2x = 4y = 12 [and evaluate] 3x = 5y = 14

These can be broken down to simplify. Take the left expression,

2x = 4y = 12

This becomes,

2x = 12 and 4y = 12

This gives,

x = 6, while y = 3

This is proved by inserting them into the equation,

2(6) = 12 and 4(3) = 12

x = 6, y = 3 [Answer Part 1]

Similarly, the second problem becomes,

3x = 5y = 14

This breaks down to,

3x = 14 and 5y = 14

x and y in this instance are fractions.

x = 14/3 and y = 14/5 [Answer Part 2]

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**Problem 7:** What is the answer to this high school math problem:

41 – 8m = 57

The problem is a simple one. The 41 should be moved to the other side of the equal sign. Whenever a value is moved to the other side, its value changes, whether from positive to negative, or from negative to positive.

Since 41 on the left side is positive, on the right side it becomes negative. The equation now becomes,

– 8m = 57 – 41

This becomes,

– 8m = 16

Dividing by – 8 gives,

m = -2 [Answer]

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**Problem 8:** Which of the two numbers is larger than the other, 7/8 or 11/12? The 7 and the 11 are called numerators, while 8 and the 12 are denominators.

The trick is to make the two denominators equal. If it is not immediately clear what the new denominator should be, the simplest and quickest way to come up with a satisfactory number is to multiply the denominators. 8 x 12 = 96. So let’s convert the two fractions into 96ths.

7/8 x 12/12 is the first calculation.

In doing this, you aren’t changing the value of your fraction, since multiplying times 12/12 is the same as multiplying times 1.

The result of the first calculation is 84/96.

Now the second number was 11/12.

Multiplying that by 8/8 gives 88/96.

So, by comparing the numbers,

11/12 is greater than 7/8 by 4/96. [Answer]

* Note:* Try to do math in your head. It’s good exercise for the brain (which is a lot like a muscle). With lots of practice, you will be able to do fairly complicated problems in your head.

It is easy to see that 7/8 is less than a whole by 1/8 (since 8/8 = 1). By the same reasoning, 11/12 is less than a whole by 1/12. Now a twelfth is smaller than an eighth, therefore 11/12 is greater than 7/8.

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