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# Three Types of Percent Problems Day 2 - PowerPoint PPT Presentation

Three Types of Percent Problems Day 2. Remember, percents are "parts of a whole". The part is the numerator and the whole is the denominator. 17% means 17 parts per 100 or We are going to solve problems involving percents . There are 3 types of problems:

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The part is the numerator and the whole is the denominator.

17% means 17 parts per 100 or

We are going to solve problems involving percents.

There are 3 types of problems:

1. Find the partвЂЇWhat number is 54% of 34?

2. Find the whole4 is 60% of what number?

3. Find the percent18 is what percent of 28?

These words have specific meanings in math.

To solve a percent problem, translate the words into an equation.

Change the following:

1. Percent into a decimal

Then, solve the equation.

Write a mathematical sentence

Write a mathematical sentence

Write a mathematical sentence

Find 12% of 70 What is 40% of 28?

You can also solve percent problems by setting up a proportion.

Since percents are parts of a whole, you can create the following proportion:

When figuring out which is the "part9quot; and which is the "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the whole is with the word "of9quot; and the part is with the word "is9quot;.

Proportion Method "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

1. Set up the proportion as shown.

2. Substitute given values into the proportion.

3. Solve the proportion.

Note: You can use this box to solve many problems

Note: Try to find the numbers that are attached to the

words/symbols: is, of, or percent.

Example: What is 25% of 400? "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

1. Set up the proportion.

Click on each box to see if you substituted correctly.

What is 25% of 400?

Example: What is 32% of 300? "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

1. Set up the proportion.

Click on each box to see if you substituted correctly.

What is 32% of 300?

Try it: What is 20% of 180? "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

1. Set up the proportion.

12 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

13 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

What is 15% of 90?

14 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

Find the greater value.

15 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

Find the greater value.

Identify any values that are equal. "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

What is 40% of 80?

Finding the Whole. "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

Remember, you can solve this by: "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

1. Translating into an equation

2. Setting up a proportion

40% of what number is 50?

Try This: "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

100 is 20% of what number?

17 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

56 is 70% of what?

18 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

12% of what number is 6?

19 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

65% of what number is 10?

27 is 150% of what number? "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

21 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

1% of what number is 12?

Finding the Percent. "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

Remember, you can solve this by: "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

1. Translating into an equation

2. Setting up a proportion

What percent of 80 is 24?

60 is what percent of 15? "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

22 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

What percent of 3 is 12?

23 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

30 is what percent of 36?

24 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

What percent of 18 is 180?

25 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

2 is what percent of 1?

26 "whole9quot;, remember that you take a percent of the whole and вЂЁthe answer is the part. In other words, the

# what is 3 percent of 300

* Percent - a special type of fraction *

These expressions tell us what portion of the square is coloured orange.

The word percent come from the expression 'per cent' and literally means 'a part of one hundred'. A percent is a part, or fraction, out of 100. For example:

We can see that to write a percent as a fraction we express the percent as a fraction with a denominator of 100. We may then be able to simplify the fraction further.

To express a fraction as a percent we must first convert the fraction into hundredths (in simple cases we can do this by using equivalent fractions) and then replace '/100' by the percent '%' sign.

We can see that we express a percent as a decimal by dividing by 100.

To express a decimal as a percent we multiply the decimal number by 100.

**Some percents expressed as fractions and decimals**

*Example 1: 30 out of 50 apples in a box are too bruised to sell. What percent of apples cannot be sold?*

30 out of 50 apples are bruised. To represent 30/50 as a percent we need to find out how many apples out of 100 are bruised.

We could also say that,

0.6 of the apples are bruised.

*Example 2: Ryan spent 25 minutes in the bank, 11 minutes of which was spent waiting in a queue. What percent of time did he spend waiting in the queue?*

Ryan spent 11 minutes out of 25 minutes waiting in a queue. To turn this into a percent we are asking, 11 out of 25 minutes equals how many minutes out of 100 minutes?

We can see that 11 mins out of 25 mins equals 44 mins out of 100 mins by equivalent fractions (because we know 25 x 4 = 100) .

We can say that Ryan spent 44%, 0.44 or 11/25 of his time in the bank waiting in a queue.

*Example 3: What percent is 7 cm of 20 cm?*

To find out what percent 7 out of 20 is, we need to ask: 7 out of 20 is how many out of 100?

5 groups of 20 make 100, so 7 out of 20 is 35 out of 100 (5 x 7 out of 5 x 20).

Therefore 7/20 equals 35%, or 0.35 if we represent it as a decimal.

**Dual-scale number line model**

We can use the dual-scale number line, also called the proportional number line, to illustrate example 1 from above.

The left side of the number line below has a percent scale. The right side of the number line has a number scale. We can label each scale using the information we are given in the problem.

We know that there are 50 apples in total, ie. 50 apples equals 100% of the apples. We know that 30 out of the 50 apples are bruised and we need to find what percent this is.

In more complicated problems this dual-scale number line is a good way of organising the information we are given and to work out what information we need to find.

Once we have represented the problem in this way we can write a proportion equation directly from the number line.

By equivalent fractions we know that 30/50 = 60/100.

(Or we might have just noticed that it is a 'multiply by 2' relationship, so 30 x 2 = 60)

Therefore 60% of apples are too bruised to sell.

The dual-scale number line model is discussed further in the other pages of the Percent, Ratio and Rates topic.

**Elastic tape measure model**

The tape measure model is a good linear model of percent. Teachers can easily make these models using a ruler, such as a 1 metre ruler, and elastic. The elastic needs to be marked with a percent scale. It can then be stretched to the desired length.

*For example, what is 60% of 50?*

To find the answer we line up the zeros of the ruler and the elastic. We then stretch the elastic so that 100% lines up with the whole amount, which in this case is 50. We then look for 60% on the elastic and read the corresponding amount on the ruler. We can see below that 60% of 50 is 30.

The intention is NOT to use this model accurately. It is a good way of showing that percent always involves a proportional comparison of something to 100.

By manipulating the tape measure, this model can be used for the 3 types of percent problems, discussed in Percent Examples. Examples of which are,

What is 20% of 50?

What percent is 10 of 50?

30% of what number is 15?

(Note: for a lesson, a teacher will need elastics tape measures of various lengths, because the elastic can only be stretched - it cannot be shrunk).

(This elastic tape measure model was developed by J. H. Weibe)

The ratio of 1 : 3 tells us the ratio of shaded : unshaded

The ratio of 3 : 1 tells us the ratio of unshaded : shaded

The ratio of 1 : 4 tells us the ratio of shaded : whole

A ratio is another way of comparing quantities. Each quantity must be measured in the same units. An advantage of ratios is that we can compare several things at once.

(every 1 cm on the map represents 10000 cm on the ground, every inch on the map represents 10000 inches on the ground)

**ratio of blue to white paint is 1 : 4**

(for every 1 litre of blue paint there are 4 litres of white paint; for every cup of blue paint there are 4 cups of white paint, i.e. 4 times as much white)

**ratio of gears on a bicycle 8 : 16 : 24**

(cog size increases in the proportion of 1:2:3:4 etc)

**ratio of number of girls to number of boys in class is 5 : 2**

(for every 5 girls in class there are 2 boys)

Although ratios must have each quantity measured in the same units, the units are not fixed . 1 litre of blue paint to 4 litres of the white paint represents the same ratio as 1 tin of blue paint to 4 tins of white paint, or 1 bucket of blue paint to 4 buckets of white paint. This fact makes ratios very versatile to use in everyday situations.

The order in which a ratio is written is very important. If we say the ratio of the number of girls to the number of boys is 5:2 this is very different to saying the ratio of the number of girls to the number of boys is 2:5.

A ratio can be written in different ways;

- in words - the ratio of the number of girls to the number of boys is 5 to 2, and this is the way we say it

- using a colon - number of girls : number of boys = 5 : 2

*Example 4:* *Let's say I want to make the paint colour 'sky blue' and I know that the way to do this is to mix 1 part blue with 4 parts white. This means there is a ratio of blue to white of 1:4. In this case 1 litre of blue to 4 litres of white, making 5 litres of sky blue paint.*

To make double the quantity of paint I can mix the blue to white as a ratio of 2:8. This will make the same colour. The ratios of 1:4 and 2:8 are equivalent, and worked out in the same way as equivalent fractions. We multiply each part of the original ratio by the same number and we can find equivalent ratios.

**Sharing quantities in a given ratio**

*Example 5:* *We have a small inheritance of $15000 to be shared among 3 people in the ratio of 2 : 2 : 1, how much does each person receive?*

The ratio of 2 : 2 : 1 means that the inheritance is divided into 5 portions - two people each receive 2 portions and one person receives 1 portion.

$15000 divided by 5 - each portion is worth $3000

* Relationships - decimal fractions, common fractions, percent and ratio*

**We can use examples to illustrate the relationships between decimal fractions, common fractions, percent and ratio.**

*Example 6: An inheritance of $15000 is to be distributed among 3 people in the ratio of 2 : 2 : 1. ($15000 will be divided into 5 portions)*

*Example 7: Ratios and fractional parts.*

* A litre of mixed cordial requires 250 mls of cordial and 750 mls of water. How can we represent this as a ratio and a fraction?*

*The ratio of cordial to water is 250 : 750 or 1 : 3. One part cordial to 3 parts water.*

* In fraction terms, the 1000 mls of mixed cordial is 250/1000 (1/4) cordial and 750/1000 (3/4) water.*

*Example 8: Ratios and fractional parts.*

* A group of 100 people is made up of 60 males and 40 females. How can we represent this as a ratio and a fraction?*

*The ratio of males to females is 60 : 40 or 6 : 4. This means that overall there is a higher proportion of males in the group, and for every 6 males there are 4 females.*

*So we can also say that 6 out of every 10 people are males and 4 out of every 10 people are females.*

*Whereas fractions only enable us to represent the part to whole relationship (in this case, males/people and females/people), different aspects of the relationships between quantities (people) can be shown using ratios. For example,*

*The 3 ratios that represent the relationships of males and females in this group of people are:*

*- the ratio of 6 males to 10 people can be represented as 6 : 10*

*Which ratio we choose depends on we want to say.*

* The number of males to the number of females is 6 : 4*

* The number of males to the number of people is 6 : 10*

* The number of females to the number of people is 4 : 10*

*In the early stages of introducing ratio at a primary level we generally discuss ratios in terms of a part to part or, quantity to quantity, comparison. At this stage part to whole relationships are often better represented by fractions or percents with which students already have some experience. When part to whole ratios are introduced care must be taken to ensure students clearly understand what is being represented. *

*We use rates when we are measuring one quantity or amount in relation to another quantity or amount. We use them to compare how quantities change, usually over a period of time. A significant difference between rates and ratios is that when we are forming rates, each quantity is measured in different units to form new composite units .*

*For example, let's say you are travelling at a rate of 60 kilometres per hour (km/h). Here we are measuring kilometres in relation to hours and the rate unit becomes ' kilometres per hour ', often written as km/h.*

*Other examples of rates are, an athlete running at 10 metres per second (m/s), and a factory using water at a rate of 450 litres per hour (l/h).*