# FIGURE 12.5.1 - Mathematics

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FIGURE 12.5.1 - Mathematics

The compression ratio of an engine is a very important element in engine performance. The compression ratio is the ratio between two elements: the gas volume in the cylinder with the piston at its highest point (top dead center of the stroke, TDC), and the gas volume with the piston at its lowest point (bottom dead center of the stroke, BDC).

There are two ways of calculating the compression ratio for an engine. Firstly, you can make a math calculation as accurately as possible, or secondly - the more popular method - uses an empty spark plug socket with a pressure gauge inserted.

We're looking at the first of these ways here. This technique is suited to someone who is in the middle of putting an engine together and has the right tools, or someone whose engine is already in pieces.

Compression Ratio: to 1

## Brief review on how to find the slope and the slope-intercept form without a slope calculator.

If you want to find the slope using the two points (2,9) and (12,3), you can use the following formula:

The slope calculator will also show that the slope-intercept form is y =  -0.6x + 10.2

Here is how to find the slope-intercept form.

The slope-intercept form is y = mx + b

Substitute either (2,9) or (12,3) for (x,y) in y = -0.6x + b

Now, enter (2,9) and (12,3) in the slope calculator above and you will see that you will get the same answer.

You can enter (2,9) in either the two boxes on the left or the two boxes on the right. Similarly,you can enter (12,3) in either the two boxes on the left or the two boxes on the right. You will get the same answer!

Buy a comprehensive geometric formulas ebook . All geometric formulas are explained with well selected word problems so you can master geometry.

## Dividing Decimals&mdashMental Math

This is a complete lesson for 5th/6th grade with instruction and exercises, teaching students how to divide decimals using mental math (based on number sense). It starts out with some sharing divisions, then explains the basic strategy for those. Students also divide decimals with "measurement division", such as 0.45 ÷ 0.05, where we think how many times the divisor goes into the dividend. The lesson has pattern exercises, word problems, a cross-number puzzle, and more.

1. First shade the parts. Then divide and write a division sentence.

Divide it into 3 parts.

Divide it into 2 parts.

Divide it into 3 parts.

Divide it into 4 parts.

Divide it into 10 parts.

Divide it into 10 parts.

A decimal divided by a whole number

You can think of multiplication &ldquobackwards.&rdquo To solve 4.5 ÷ 5, think: What number multiplied
by 5 will give me 4.5?
Or, _____ × 5 = 4.5. The answer is 0.9.

Or, think of “bananas” divided among a group of people. The only thing is, this time
the “bananas” are tenths, hundredths, or thousandths!

For example, 0.035 ÷ 5 is “35 thousandths divided by 5”. Replace the thousandths by bananas
for a moment: “35 bananas divided by 5. equals 7 bananas.” The answer to the original
problem is 7 thousandths, or 0.007.

Another example: 0.12 ÷ 4 is “12 hundredths divided by 4”. This is essentially the division
problem “12 divided by 4”, however, in terms of hundredths. The answer is 3 hundredths or 0.03.

2. Write the division problems with numbers, and solve.

 a. 9 tenths divided by 3 equals . _______ ÷ ____ = _______ b. 72 thousandths divided by 9 equals . _______ ÷ ____ = _______ c. 54 hundredths divided by 6 equals . _______ ÷ ____ = _______ d. 240 thousandths divided by 60 equals . _______ ÷ ____ = _______ e. 122 hundredths divided by 2 equals . _______ ÷ ____ = _______

3. Divide. Think of dividing “bananas”: how many tenths, hundredths, or thousandths you are dividing.
Or, think of multiplication backwards.

4. Divide. Tag a zero or zeros on the dividend.

5. Jane shared $2.00 equally among five friends. How much did each one get? 6. If each heartbeat takes 0.8 seconds, how long do five heartbeats take? 7. Write two division problems and two multiplication problems with the same numbers&mdasha fact family! Example 2. Mom cut 0.4-meter pieces from a 1.2-meter piece of material. How many pieces did she get? Think, &ldquoHow many times does 0.4 go into 1.2?&rdquo The answer is of course easy: 3 times. We can also write a division from this situation: 1.2 ÷ 0.4 = 3. 8. Divide. Think: how many times does the divisor go into the dividend? e. 0.006 ÷ 0.002 = ______ 9. Write a division sentence for each problem, and solve. a. How many 0.3 m pieces do you get from 1.8 m of cloth? _______ ÷ _____ = ________ b. How many 0.7 m pieces do you get from 4.2 m of wood? _______ ÷ _____ = ________ c. How many 0.05 m pieces do you get from 0.25 m of string? _______ ÷ _____ = ________ First, tag a zero on 0.72 so that it also has three decimals, just like 0.008 has three decimals. Now we get: 0.720 ÷ 0.008 = ? Now think, “How many times does 8 thousandths fit into 720 thousandths?” This is the same as asking, “How many times does 8 fit into 720?” The answer: 90 times. So, 0.720 ÷ 0.008 = 90 (not 0.90 or 0.090 just plain 90) . 10. Divide. You may need to tag a zero or zeros on the dividend so that both numbers have the same amount of decimal digits. Then think: How many times does the divisor go into the dividend? a. 0.2 0 ÷ 0.05 = _______ 11. The asphalt crew does a 1.2-mile stretch of road each day. a. How many days does it take for them to cover a distance of 6 miles? b. How many days does it take for them to cover a distance of 60 miles? 12. Jack has$1.45 in nickels in his pocket.

a. How many nickels does Jack have?

b. If you have not already, write a decimal division to match the problem.

13. How many 0.04-meter sticks can you cut from a 0.20-meter board?
Write a decimal division to match the problem.

15 cm + 3 cm + 1.5 cm = 19.5 cm

15. Write a single expression (number sentence with several operations) to match this problem. Solve.

How much is left from 5 meters of material
after you cut off four 0.6 meter pieces?

16. Joe has 0.85 kg of meat. How many 0.3 kg servings can he get from that?

Also, &ldquoconvert&rdquo this problem into grams, remembering that 1 kg has 1,000 grams.

17. Divide and place the answers in the cross-number puzzle.

18. Figure out the pattern and continue it for at least two more problems.

## Tuesday, November 28, 2006

### 12.6.1 "The Law of Large Numbers" (Up to middle of page 888)

Difficulty: Markov's inequality and Chebyshev's Inequality are given to us in the chapter. However, I do not understand how these inequalities may be applied to finding the true probability. In addition, for Chebyshev's Inequality, does this work for all random variables or only certain types? (for example, exponentially distributed RVs, etc.).

Reflective: It is logical that a larger sample space would provide more accurate results (the Law of Large Numbers), since results would not be dependent on a few objects in the sample space which may not give an accurate representation. The law of large numbers exist in everyday life, such as surveys and polls. To gain accurate results in polls, it is a better idea to poll 100 people rather than 10 people since few extreme individuals may alter the general results in polls with small sample sizes.

## Comparing decimals

This is a complete lesson with instruction and varied exercises about comparing decimals with 1 or 2 decimal digits. A student with a common misconception will say that 0.16 is more than 0.4, thinking of the decimal digits as "plain numbers." We can use place value charts to combat this misconception.

Review. Which is greater, 4506 or 4606? How do you know?
Which is greater, 4512 or 4562? How can you tell?

Which is greater, 4603 or 4478? How can you tell?

Challenge. How well can you do on comparing decimal numbers?

Decimals are compared in exactly the same way as other numbers: by comparing the different place values from left to right. To help in that, you can write the two numbers into the place value tables on top of each other. Then compare the different place values in the two numbers from left to right, starting from the largest place.

2. Write the following numbers in order. Remember: It is easier to compare if the numbers have the same amount of decimals. You can also use the number line above to help.

5.01 5.3 5.03 5.19 5.1 4.9 5.24 4.92 5.15 5.5 4.8

3. Compare and write <, >, or = . Use the place value tables if you need to.

5. Choose the largest number.

 a. 7.85 7.8 7.5 b. 15.4 15.44 15.04 c. 2.37 2.77 2.7 d. 3.09 3.9 3.91 e. 0.30 0.36 0.3 f. 0.8 0.48 0.79

6. a. Write these numbers from smallest to greatest:
1.4 1.34 1.44 1.5 1.3 1.30 1.28 1.49

b. Draw a number line from 1.2 till 1.5 with tick marks at every hundredth. Mark the numbers
from a. on it, and thus check your work.

7. Write the numbers in order from smallest to greatest.
0.9 0.67 0.04 0.05 0.90 0.03 0.34 0.4 0.2 0.21

8. Give an example of two decimal numbers where
a. the number with more decimal digits is smaller than the other
b. the number with more decimal digits is bigger than the other
c. the number with more decimal digits is equal to the other

## Diluting Stock Solutions by Percentage

The dilution equation works even when you don't have a molarity associated with the stock. Let's say someone gives you a 10% stock solution of sodium azide, and you need to make 500 mL of a 0.1% working solution. You can use the same equation to do so as shown here:

This means you will need 5 mL of the 10% stock solution and 495 mL of diluent.

As long as you have three pieces of information to plug into the dilution equation you will be able to solve for the last unknown. Now you have a quick way to calculate dilutions when you have a stock solution.

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