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Calculating Rate
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Skills Application: Calculating Rate
Step-by-Step: Calculating Rate
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On 14 October 2012 Felix Baumgartner landed in New Mexico after jumping from a world-record altitude. He ascended to the edge of space and also broke the freefall speed record. His success is no coincidence. His rate of decent was carefully calculated beforehand. The jump altitude was 38,969.3 metres (127,852 feet). He reached supersonic speed in just 40 seconds, still accelerating in near-vacuum. After reaching terminal velocity (Mach 1.1) and about 2 minutes of freefall, his rate of descent had to be changed very carefully. Friction with air molecules began to slow him down acting as a brake, which shook him violently and heated his space suit up. Within 3.5 minutes this friction induced a dangerous spin, from which Baumgartner recovered. His parachutes were opened after 4.5 minutes in freefall at about 1500m (5000 feet) above terrain. From this altitude his rate of descend was a very casual 1000 feet per minute. His descent took a total of about 9.5 minutes, he landed safely.

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Understanding Percent Problems

All calculations involving percent can fall into one of the three basic types of problems. These three are related to all percent problems that arise in business, technology, or the trades. By learning these three types of problems you will be able to solve any percent problem, and we examine each of the three types of problems. All percent problems involve three quantities:
• the base ( B ) or whole or total amount, a standard used for comparison
• the portion ( P ) or part being compared with the base
• the rate ( R ) or percent, a percent number
Identify the Parameters:

To solve any percent problem, you need to identify which of the quantities given in the problem is P, which is B, and which is R. The percent number R is easy to identify because it always has the % symbol attached to it. If you have trouble distinguishing the part P from the base B, notice that B usually follows the word “of” and P usually follows the word “is”.

Example: What is 20% of 280?

• is indicates the portion (or percent) P to be found.
• % indicate the percent rate R, here R = 30%
• of indicates the base B which is equal to 280.
Find the Missing Percentage from Two Given Numbers

The next three sample sentences are all forms of the same problem. In each sentence the percent R is unknown. We know this because neither of the other two numbers has a % symbol attached.

•  7 is what percent of 12?
•  Find what percent 7 is of 12.
•  What percent of 12 is 7?

Finding the Missing Percent

To find the missing percent when given two numbers, do the following:

1. Write a fraction with the two numbers. The number after the word “of” is always the denominator, and the other number is the numerator.
2. Simplify the fraction (if possible).
3. Change the fraction to a decimal.
4. Express the decimal as a percent.

EXAMPLE

Now lets solve this problem: What percent of 12 is 7?

SOLUTION

1. Write a fraction with the two numbers, the bigger is usually the base or denominator:

$\frac{7}{12}$

2. Simplify the fraction (if possible):

$\frac{7}{12}$
is at its lowest terms since 7 is a prime number.

3. Change the fraction to a decimal:
7÷12 = 0.58

Express the decimal as a percent:

0.58 = 58%

We can say that 58% of 12 is 7. When you find the percentage, remember to include the % symbol with your answer.

Check out the ADVANCED section. The exercises will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. It is important to try alternative calculations or approaches, tracking changes to units of measure, visualizing to cross-link the concepts of percentage and numbers. If you spend 10 – 20 min working with the different ideas in the ADVANCED page, youll remember key formulas easier and youll learn new skills faster. ### A Short Review

Example:

What percent of 30 is 10?

SOLUTION

1. Write a fraction with the two numbers:
$\frac{10}{30}$
2. Simplify the fraction (if possible):
$\frac{10}{30} = \frac{1}{3}$
3. Change the fraction to a decimal:
1÷3 = 0.3333

Express the decimal as a percent:

0.3333 = 33.33%

We can say that 33.33% of 30 is 10. When you find the percentage, remember to include the % symbol with your answer. ### Worksheet: Level 1

The operations used are clearly specified. Only one type of mathematical operation is used in a task.

### Worksheet: Level 1 Sample Question

Example:
In the fraction
$\frac{8}{20}$
which number is the part or portion, and which is the whole or base?

### Worksheet: Level 1 Answer Key

8 = Part
20 = Whole ### Worksheet: Level 2

Tasks involve one or two types of mathematical operation. Few steps of calculation are required.

### Worksheet: Level 2 Sample Question

Example:

Calculate the percent rate in the following word problems:

24 outlets are installed out of 751. What % of the job is done? Round the answer to 1 decimal digit.

### Worksheet: Level 2 Sample Answer

24 ÷ 751 × 100 = 3.2%

$\frac{24}{751}$
= 24 ÷ 751

24 ÷ 751 = 0.0319

0.0319 x 100 = 3.2% ### Worksheet: Level 3

Tasks require a combination of operations. Several steps of calculation are required.

### Worksheet: Level 3 Sample Question

Example:

Calculate the percent rate of the following amounts:

20 is what % of 40?

### Worksheet: Level 3 Sample Answer

$\frac{20}{40}$
= 20 ÷ 40

20 ÷ 40 = 0.5

0.5 × 100 = 50 → 50% Tasks involve multiple steps of calculation. Advanced mathematical techniques may be required.

### Worksheet: Advanced Level Sample Question

Calculate what percent (rate) is 40 of 250? Or: 40 out of 250 is what %?

Another way to get the answer, using division and multiplication:

1.  First off, recognize that the percent rate can be calculated given a portion and base.

2.  Recognize that 250 is the base, and a portion of it is 40. The biggest number without the % sign is usually the base.

3.  Set up the problem:

portion ÷ base x 100 = rate (%)

4.  Calculate:   40  ÷ 250  x  100 =  16

This 100 comes from the fact that percents always have an invisible “out of 100” concept with them. The % sign is actually short for “out of 100”.

5. Take a pencil and write down  16

6. Write the % sign after the 16 Now youre done with the math.

7. The last step: check your work, make sure everything was copied and written correctly then determine that the correct way to write the answer is 16% 40 is 16% of 250 Compare the animated calculation with the one in the additional information.

In the second approach, the steps were:    40  ÷ 250  x  100

and in the first approach the sequence was:    40  ÷ 250  followed by the “decimal result x 100”.

Either way, the answer is 16. These 2 calculations are also the same, you can enter into the calculator either

40  ÷ 250  x  100 =       to get 16 or

press more buttons and enter:  40  ÷ 250  =  x  100 =  to get 16.

Compare the 2 explanations for the calculations.

In the second approach, the process was:     “portion ÷ base x 100 = rate (%)”

and in the first approach we have:

“numerator ÷ denominator = decimal”       then,     “decimal x 100 = percent”.

Notice that portion ÷ base = rate, which is 0.16, which is the same as numerator ÷ denominator. Essentially this decimal 0.16 is rate, but is not in percent format. It needs a “x 100” to get percent. So, in one approach the explanation is that this decimal (0.16) needs to be multiplied by 100 to get a percent, and in the other we get rate (%)” at the end of the equation, not just rate”.

Memorize the lines of explanations that makes the most sense to you. Either way, you will be entering the same numbers in the calculator in the same order, getting the same final answer.