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Calculating Base
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Skills Application: Calculating Base
Step-by-Step: Calculating Base
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King Hiero appointed a goldsmith to fashion a crown out of pure gold; after hearing rumors that the crown was not pure gold the King ordered the mathematician Archimedes to verify the gold content.

One day when taking a bath, Archimedes discovered a way to solve the problem; he noticed that when he was lowering himself into the bathtub the more his body sank into the water the more water ran out over the sides of the tub, “Eureka, ‘I have found it!”

2.5 ÷ 20 x 100 = 12.5

Archimedes knew that gold was denser than silver and gold would displace less water than silver. After some experiments he concluded that indeed the crown was not pure gold.

With 24kt pure gold as the base metal, by adding 25% of silver you get 18kt gold If you add 50% silver to the base gold you get 12kt gold and so on…

Running with Pi

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3. Our Training Worksheets

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.
Words in base problems

This third type of percent problem requires that you find the whole (or base) when the portion (or part) and the rate (in percent) are given. Problems of this kind can be worded as:

•  18 is 15% of what number?
•  Find a number such that 15% of it is 18.
•  15% of what number is 18?

Finding the base

Finding the base when percent and portion are given is done in 2 steps:

1. Change the percent to a decimal
2. Divide the portion by this decimal.

EXAMPLE

15% of what number is 18?

SOLUTION

In this problem the percent is the number with the percent symbol (15%). The other number is the portion (18).

1. Change the percent into decimal:  15% = 0.15
2. Divide the portion by the decimal:  18 ÷ 0.15 = 120

The answer is: 15% of 120 is 18.

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 20 – 40 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:
50 is 25% of what number?
1. Change the percent into decimal: 25% = 0.25
2. Divide the portion by the decimal: 50 ÷ 0.25 = 200

The answer is: 25% of 200 is 50

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:

Calculate the base of the following amounts:

10% of what amount is 231?

$\text{10% = }\frac{10}{100}$

$\frac{10}{100}$
= 10 ÷ 100

10 ÷ 100 = 0.1

231 ÷ 0.1 = 2310

Worksheet: Level 2

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

Worksheet: Level 2 Sample Question

An outdoor swimming pool can lose 6% of its water, or 1500 L a day in August.

How much water does the pool normally contain?

$\text{10% = }\frac{6}{100}$

$\frac{6}{100}$
= 10 ÷ 100

6 ÷ 100 = 0.06

1500 ÷ 0.06 = 25, 000 L

Tasks involve multiple steps of calculation. Advanced mathematical techniques may be required.

Math in real life does not often happen in a ready-to-calculate pre-printed format. The math is very often hidden in word  problems. The following word  problems are all based on the math in this lesson. Here  is how to deal  with word  problems:

1. Read the word problem and  write down the numbers from the word problems in the order they appear with their units  of measure. For example:

5 m
18%
72 hrs
21 lbs 2 oz

2. Write the  meaning of or label  the  copied out  numbers with their  units  of measure.  For example:

Part  = 3.8 carton
Width = 5 m or w = 5 m
Taxes = 18%  or VAT = 18%
Work in Pay Period = 72 hrs or Labour = 72 hrs
Weight  = 21 lbs 2 oz or w = 21 lbs 2 oz

3. Identify what  exactly needs to be calculated and  write it down. For example:

Volume = ? or V = ?
Density = ? or ρ = ?
Angle = ? or = ?
Length = ? or L = ?

4. Read all the words of a word  problem. The math is hidden, only the words indicate math.

For example:

- Stretching out, adding to, making more, growing, getting bigger, heavier, and building anew almost always means addition
- cutting, shortening, getting less, loss or losing all imply subtraction
- "of” or "at" following or near numbers means multiplication, many  pieces of uniform  units  (boxes, crates, loads, lifts, nights, hours) can  also  be multiplied
- chopping, sharing, splitting, slicing, or breaking up unto identical groups means division, so does the word "per", such as per student, per box, per room.

5. Based on the clues in the text, choose the applicable math procedure that  will answer the problem. In this  case, you don’t  have to do this.  The word problems in this lesson are solved with math shown in this lesson.
6. Recall steps of the calculation to get the answer or review and follow the steps in a sample calculation.
7. Calculate the final answer and write it down with its correct unit of measure.

Tasks involve multiple steps of calculation. Advanced mathematical techniques may be required.

1. Compare the 2 calculations done on a “calculator with a % button” in Episodes 6 and 8. Only portion and base calculations use the % button. Calculating rate (episode 07) cant possibly use the % button. Why?
1. Look at & compare the portion or base calculations done on a “calculator with a % button” and the steps that don’t use the % button. The % button is a convenience shortcut for 1 well-known math operation. What is this exact built-in math operation the calculator does when the % button is pressed?
1. Addition and subtraction are reverses of each other. A whole number (1) can be subtracted from a number (4) and added back to the result (3) to get the same number we started with (4):

4     –    1    =     3          , then          3     +    1     =     4

It does not work out the same way with percents:

$4.00 – 25% =$3.00     , but       $3.00 + 25% =$3.75 only.

Why does adding and subtracting with percents not work out like it does with whole numbers?

4. Look at Episode 1, adding and subtracting percents. How does that addition or subtraction differ from the above example?

5. To solve % word problems efficiently, you need to be able to:
a) Identify which calculation to do (based on word clues) and
b) You have to remember the formula to use accordingly.

Contrary to popular belief, calculators are not much help as they dont do either one of these 2 things. Calculators are limited tools. They dont do the thinking or the math. They only calculate.

To help you get started with memorizing the formulas from Episodes 7, 8, and 6, print them out for yourself, study & compare.
portion ÷ base x 100 = rate          portion ÷ percent x 100 = base                   percent ÷ 100 x base = portion

The full words in the formulas can be replaced by a single letter. This is how they look now:

$\frac{P}{B}\text{ = R}$
(Episode 7)
$\frac{P}{R}\text{ = B}$
(Episode 8)                          B x R = P   (Episode 6)

a)       The following visual of a triangle may help with memorizing % formulas and their relationships:

Portion, base and rate can combine in 3 ways (see the 3 formulas above):
If Portion is known, it can only be divided, either by Base or Rate.
(P is always over B or R, “over” implies fractions, and all fractions are divisions)
If Base and Rate are known, they are to be multiplied.
(In formulas, amounts for multiplication are always written side-by-side)

b)      Some people like the 3 quantities arranged like fractions:

$\frac{P}{B}\text{ = R}$

$\frac{P}{R}\text{ = B}$
B x R = P
These letters can be replaced by numbers to see an interesting relationship.
Portion (P) can be replaced by 1, base (B) can be 4, and (R) rate can be replaced by 0.25
$\frac{1}{\color{Blue}4}$
= 0.25
$\frac{1}{\color{Red}{0.25}}$
= 4                                                4 x 0.25 = 1

Notice that in fractions the 4 and 0.25 (the denominator and the result) can trade places. This fraction fact may help memorize the formulas knowing that if portion (P) is given a division must follow by either the base or rate to calculate the missing quantity.

A formula is a symbolic short-hand way to write calculation procedures. Use a memory jogger thats easiest to remember for you.

1. Because percent is the final answer, it can not be entered
2. ÷ 100 =
3. Because there is a hidden multiplication with the above subtractions:
4 – (4 x 25/100) = 3,        and        3 + (3 x 25/100) = 3.75
The bases of the different percentage calculations are different, 4 in the first case, then 3 in the second.)

4. They dont involve a base

Infosheet: Another Way to Get the Answer (using division and multiplication)

Calculate: 20% of what amount is 2.5 hrs?

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

1. First off, recognize that “base” can be calculated given a percent rate and portion. 20% is the rate, 2.5 is a portion of a bigger amount we dont have yet.
2. Set up the problem:

portion ÷ percent x 100 = base

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”.

1. Calculate:   2.5   ÷    20       x  100 = 12.5
2. Take a pencil and write down 12.5
3. Write hrs after the 12.5 Now you`re done with the math.
4. 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 12.5 hrs

20% of 12.5 is = 2.5

Infosheet: Another Way to Get the Answer (using a calculator)

Another way to get the answer, using a calculator with a % button:

1. Enter 2.5
2. Press ÷
3. Enter 20
4. Press %

The number on the display changes to 12.5

20% of 12.5 is = 2.5