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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsCommercial ArithmeticsComparing Quantities

### Introduction to Percentages

A ratio is represented with two numbers, eg: 3:4. Comparing two different ratios involve some arithmetics. eg: 4:5 and 3:4, which one is larger in ratio?

To simplify such comparison, the second term is standardized to a value, say 100, then the given ratios are 80:100 and 75:100. It is easier to compare these ratios.

Such standard representation is simplified by dropping the known number 100. and the simplified form is "percentage". eg: 80% and 75%.

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Consider a fruit basket with apples, oranges, and bananas

•  The ratio of apples to bananas is 1:4

•  The ratio of oranges to bananas is 2:9

Can one figure out if apples or oranges are more in the basket?

• it is not evident from the given two ratios
• it is not evident from the given two ratios
• ratio of apples to oranges is 1:2 as per the given two ratios

The answer is "it is not evident from the given two ratios". The ratios are to be converted to similar ratios. This is explained in the next page.

Consider a fruit basket with apples, oranges, and bananas

•  The ratio of apples to bananas is 1:4. This means that, there is 1 apple for every 4 bananas. Or 2 apples for every 8 bananas.

•  The ratio of oranges to bananas is 2:9. This means that, there are 2 oranges for every 9 bananas. Or 4 oranges for every 18 bananas.

The two ratios can be converted as
Ratio of apples to bananas is 9:36 and ratio of oranges to bananas is 8:36. In this, 9 is greater than 8, so number of apples is greater than number of oranges.

The comparison of 1:4 and 2:9 is harder as the second term is not identical. These are converted to 9:36 and 8:36.

To simplify such comparison, which of the following is a good idea?

• nothing can be done to simplify this problem
• the second term can be standardized to one value like 100 and all ratios are specified as x:100
• the second term can be standardized to one value like 100 and all ratios are specified as x:100

The answer is "the second term can be standardized to one value like 100 and all ratios are specified as x:100".

To simplify understanding and using comparison of quantities, the second term is standardized to 100 and the ratio is given as "percentage"

Ratio 1:4 equals 25:100 which is given as a percentage 25%.

Ratio 2:9 equals 22.22 : 100, which is given as a percentage 22.22%

Which of the following is a meaning for the word "percent"?

• for every hundred
• for every hundred
• measure of area of land

The answer is "for every hundred". "per-cent" meaning "for every-hundred"

What is the term used to refer "for every hundred"?

• Pronunciation : Say the answer once
Spelling: Write the answer once

The answer is "percent".

Number of apples is 25% of number of oranges. How is the % pronounced?

• percent
• percent
• o by o

The answer is "percent"

How is the given ratio 25% pronounced?

• Pronunciation : Say the answer once
Spelling: Write the answer once

The answer is "25 percent".

Consider 4 apples and 16 oranges. Which of the following is the number of apples given as percentage of number of oranges?

• Number of apples is 25% of number of oranges
• Number of apples is 25% of number of oranges
• Number of apples is 20% of the number of fruits

The answer is "the Number of apples is 25% of the number of oranges"

Consider 4 apples and 16 oranges. Which of the following is the number of apples given as percentage of number of fruits?

• Number of apples is 25% of number of oranges
• Number of apples is 20% of the number of fruits
• Number of apples is 20% of the number of fruits

The answer is "Number of apples is 20% of the number of fruits".

Consider 4 apples and 16 oranges.

•  Number of apples as percent of number of oranges is 4/16 xx 100 which is equivalently given as Number of apples is 25% of number of oranges.

•  Number of apples as Percent of the number of fruits is 4/(4+16) xx 100 which is equivalently given as number of apples is 20% of the number of fruits.

Note that the percentage is given in two forms. First is the percentage of one quantity with reference to another quantity. Second is the percentage of one quantity with reference to the whole.

Students are reminded to note the context in which a percentage is defined.

Percentage : Comparison of one quantity to another OR specification of one quantity in the whole given as a number for every hundred.

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