10/8/2021

## Scenario 2 Part 1.2.1&your Digital Footprint

74

Signals and Systems: Part II / Solutions S3-3 x(- t) and x(1 - t) are as shown in Figures S3.4-2 and S3.4-3. X (-t) -12 Figure S3.4-2 x(1-t) x1-0. Scenario 7.2: The production function for earthquake detectors (Q) is given as follows: Q = 4K1/2L1/2 where K is the amount of capital employed and L is the amount of labor employed. The price of capital, PK, is \$18 and the price of labor, PL, is \$2. Refer to Scenario 7.2. Suppose that you receive an order for 60 earthquake detectors.

Percent - a special type of fraction Percent Models Ratios
Relationships:decimal fractions, common fractions, percent and ratio Rates Quick quiz

Percent - a special type of fraction

 0.25, 1/4, 25%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:

 100% =100/100 =1 = 1.0 (decimal) 50% = 50/100 = 5/10 = 1/2 = 0.5 = 0.50 (decimal) 25% = 25/100 = 5/20 = 1/4 = 0.25 (decimal) 40% = 40/100 = 4/10 = 2/5 = 0.4 (decimal) 5% = 5/100 = 1/20 = 0.05 (decimal) 0.5% = 5/1000 = 1/200 = 0.005 (decimal)

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.

For example, 75% = 75/100 = 3/4

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.

For example, 4/5 = 80/100 = 80%

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

For example,

 25% = 25/100 = 0.25 (twenty-five hundredths) 47.3 % = 47.3/100 = 0.473 (forty seven hundredths and 3 thousandths) 200% = 200/100 = 2

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

For example,

 0.108 = 0.108 x 100 = 10.8% 0.75 = .75 x 100 = 75% 1.2 = 1.2 x 100 = 120%

Some percents expressed as fractions and decimals

 = 0.125 (decimal) = 0.236 (decimal) = 0.333 (decimal, rounded to 3 decimal places) = 0.5 = 0.50 (decimal) = 0.667 (decimal, rounded to 3 decimal places) = 1.1 (decimal) = 1.5 (decimal) =2.0 (decimal)

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

 Working Out Thinking 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. By equivalent fractions we know that 30 out of 50 equals 60 out of 100, so 60% of the apples are bruised. We could also say that, 3/5 of the apples are bruised 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?

 Working Out Thinking 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?

 Working Out Thinking 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.

Percent models

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.

 Recall example 1: 30 out of 50 apples in a box are too bruised to sell. What percent of apples cannot be sold? Thinking 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.30/50 = ?/100By 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.

 1 metre rulerElastic

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?
Whatpercent 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)

Ratios

 The ratio of 1 : 3 tells us the ratio of shaded : unshadedThe ratio of 3 : 1 tells us the ratio of unshaded : shadedThe 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.

 scale on a map1:10000 (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 : 248:16:24 teeth on each cog (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

 We must always talk about ratios in context. To say or write 5 : 2, for example, has no meaning on its own.

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.

 1 blue : 4 white

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.

 1 : 4 1 : 4 1 : 4 x 2 x 3 x4 2 : 8 3 : 12 4 : 20

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?

Working OutThinking
 Person 1 Person 2 Person 3 2 : 1 (2 x \$3000) (2 x \$3000) (1 x \$3000) \$6000 \$6000 \$3000

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)

 Person 2 ratio 2 :(2 x \$3000) 1 (1 x \$3000) common fraction (of \$15000) decimal fraction 0.4 (of \$15000) 0.2 (of \$15000) percent 40 % (of \$15000)

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?

 Thinking 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. We talk about the ratio of cordial to water in many different ways: For every cup of cordial there are 3 cups of water. There is 3 times as much water as cordial. 1 out of every 4 parts of the mixed cordial is cordial. Can you think of any more ways?

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?

 Thinking 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. In fraction terms, where we are talking about the group of 100 people, 60/100 (6/10) are male and 40/100 (4/10) are female.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 - the ratio of 4 females to 10 people can be represented as 4 : 10 - for every 6 males there are 4 females can be represented as6 : 4Which 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.

Rates

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).

Some rates we commonly use are:

 km/h kilometres per hour c/L cents per litre \$/m dollars per minute c/m cents per minute

Example 9: A hose is running water at the constant rate of 100 litres an hour.
1. How many litres will run in 2 hours?
2. How many hours will it take to run 350 litres?

 Working Out Thinking 100 litres/1 hour = ? litres /2 hours200 litres will run in 2 hours. We know that 100 litres runs in 1 hour and we need to find out how many litres runs in 2 hours. We can write this as litres/hour because this is what we are trying to find out. By equivalent fractions we know that 100/1 = 200/2. Therefore 200 litres will run in 2 hours.

 Working Out Thinking 100 litres/1 hour = 350 litres /? hours350 litres/ ? hours = 100 litres /1 hour350 litres/3.5 hours = 100 litres/1 hour We know that 100 litres runs in 1 hour and we need to know how long it takes to run 350 litres. 350 litres is 3.5 times 100 litres, so it takes 3.5 times as long i.e. 3.5 hours. Therefore 350 litres will run in 3.5 hours.

Quick quiz

 1. Express the following percents as fractions and decimals: a) 95% b) 13.5% c) 42% d) 1% e) 0.1% 2. Express the following fractions as percents: a) 37/100 b) 164/100 c) 25/50 d) 14/20 e) 16/25 3. Express the following decimals as percents: a) 0.01 b) 0.83 c) 0.005 d) 1.10 e) 0.2 4. Express the following quantities as ratios: a) There were 3 boys for every 5 girls at school assembly. b) Nine people out of every ten watch television every night. c) In a class of 25 students 3 are left handed and 22 are right handed. d) To cook rice you need 1 cup of rice to 2 cups of water. e) To make ANZAC biscuits, you add the same amount of flour to sugar. 5. The following ratios of pets owned have been obtained from surveys of 5 local neighbourhoods. Find at least one equivalent ratio for each of the following: a) dogs : cats = 10 : 20 b) cats : dogs = 30 : 50 c) dogs : cats = 100 : 100 d) dogs : cats : guinea pigs = 6 : 4 : 2 e) fish : turtles = 75 : 25 6. Express each part-to-part ratio below as a common fraction, decimal fraction and a percent of the whole: a) men : women : children = 3 : 3 : 4 b) men : women : children = 11 : 4 : 5 c) adults : children = 4 : 1 d) men : women : children : pets = 5 : 3 : 6 : 1 7. Write each of these sentences as a rate: a) She ran 100 metres in 15 seconds. b) The bus travelled 850 kilometres in 10 hours. c) Ilana can type 160 words in 2 minutes. d) The factory packaged 600 packets of biscuits in 5 minutes.

Extra questions:

Shade 20% of this diagram

(a) What percentage of the following shape is shaded?

(b) What percentage is not shaded?

3. What is the ratio of pink to white in the following figure?

4. A biro is 23 cm long and a ruler is 30 cm long.

(a) What is the ratio of the biro to the ruler?

(b) What is the ratio of the ruler to the biro?

5. Complete the following table.

 Percentage Common Fraction Decimal Fraction 32% 0.06

6. A dog is three times as long as a bird. The dog is 90 cm long.

(a) How long is the bird?

## Scenario 2 Part 1.2.1&your Digital Footprint System

(b) Write a ratio to show the relationship of the dog to the bird.

7. I have a 300g bag of sugar and a half a kilo of butter.

(a) What is the ratio of the sugar to the butter?

## Scenario 2 Part 1.2.1&your Digital Footprint Worksheet

(b) What is the ratio of the butter to the sugar?

8. What two quantities are being measured when the units are km/hr?

Below are multiple fraction calculators capable of addition, subtraction, multiplication, division, simplification, and conversion between fractions and decimals. Fields above the solid black line represent the numerator, while fields below represent the denominator.

## Big Number Fraction Calculator

Use this calculator if the numerators or denominators are very big integers.

In mathematics, a fraction is a number that represents a part of a whole. It consists of a numerator and a denominator. The numerator represents the number of equal parts of a whole, while the denominator is the total number of parts that make up said whole. For example, in the fraction
 3 8
, the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator. If a person were to eat 3 slices, the remaining fraction of the pie would therefore be
 5 8
as shown in the image to the right. Note that the denominator of a fraction cannot be 0, as it would make the fraction undefined. Fractions can undergo many different operations, some of which are mentioned below.

Unlike adding and subtracting integers such as 2 and 8, fractions require a common denominator to undergo these operations. One method for finding a common denominator involves multiplying the numerators and denominators of all of the fractions involved by the product of the denominators of each fraction. Multiplying all of the denominators ensures that the new denominator is certain to be a multiple of each individual denominator. The numerators also need to be multiplied by the appropriate factors to preserve the value of the fraction as a whole. This is arguably the simplest way to ensure that the fractions have a common denominator. However, in most cases, the solutions to these equations will not appear in simplified form (the provided calculator computes the simplification automatically). Below is an example using this method.

 a b
+
 c d
=
 a×d b×d
+
 c×b d×b
=
 ad + bc bd
EX:
 3 4
+
 1 6
=
 3×6 4×6
+
 1×4 6×4
=
 22 24
=
 11 12

This process can be used for any number of fractions. Just multiply the numerators and denominators of each fraction in the problem by the product of the denominators of all the other fractions (not including its own respective denominator) in the problem.

EX:
 1 4
+
 1 6
+
 1 2
=
 1×6×2 4×6×2
+
 1×4×2 6×4×2
+
 1×4×6 2×4×6
=
 12 48
+
 8 48
+
 24 48
=
 44 48
=
 11 12

An alternative method for finding a common denominator is to determine the least common multiple (LCM) for the denominators, then add or subtract the numerators as one would an integer. Using the least common multiple can be more efficient and is more likely result in a fraction in simplified form. In the example above, the denominators were 4, 6, and 2. The least common multiple is the first shared multiple of these three numbers.

 Multiples of 2: 2, 4, 6, 8 10, 12 Multiples of 4: 4, 8, 12 Multiples of 6: 6, 12

The first multiple they all share is 12, so this is the least common multiple. To complete an addition (or subtraction) problem, multiply the numerators and denominators of each fraction in the problem by whatever value will make the denominators 12, then add the numerators.

EX:
 1 4
+
 1 6
+
 1 2
=
 1×3 4×3
+
 1×2 6×2
+
 1×6 2×6
=
 3 12
+
 2 12
+
 6 12
=
 11 12

### Subtraction:

Fraction subtraction is essentially the same as fraction addition. A common denominator is required for the operation to occur. Refer to the addition section as well as the equations below for clarification.

 a b
 c d
=
 a×d b×d
 c×b d×b
=
 ad – bc bd
EX:
 3 4
 1 6
=
 3×6 4×6
 1×4 6×4
=
 14 24
=
 7 12

### Multiplication:

Multiplying fractions is fairly straightforward. Unlike adding and subtracting, it is not necessary to compute a common denominator in order to multiply fractions. Simply, the numerators and denominators of each fraction are multiplied, and the result forms a new numerator and denominator. If possible, the solution should be simplified. Refer to the equations below for clarification.

 a b
×
 c d
=
 ac bd
EX:
 3 4
×
 1 6
=
 3 24
=
 1 8

### Division:

The process for dividing fractions is similar to that for multiplying fractions. In order to divide fractions, the fraction in the numerator is multiplied by the reciprocal of the fraction in the denominator. The reciprocal of a number a is simply
 1 a
. When a is a fraction, this essentially involves exchanging the position of the numerator and the denominator. The reciprocal of the fraction
 3 4
would therefore be
 4 3
. Refer to the equations below for clarification.
 a b
/
 c d
=
 a b
×
 d c
=
EX:
 3 4
/
 1 6
=
 3 4
×
 6 1
=
 18 4
=
 9 2

## Scenario 2 Part 1.2.1&your Digital Footprints

### Simplification:

It is often easier to work with simplified fractions. As such, fraction solutions are commonly expressed in their simplified forms.
 220 440
for example, is more cumbersome than
 1 2
. The calculator provided returns fraction inputs in both improper fraction form, as well as mixed number form. In both cases, fractions are presented in their lowest forms by dividing both numerator and denominator by their greatest common factor.

## Scenario 2 Part 1.2.1&your Digital Footprint Software

Converting from decimals to fractions is straightforward. It does however require the understanding that each decimal place to the right of the decimal point represents a power of 10; the first decimal place being 101, the second 102, the third 103, and so on. Simply determine what power of 10 the decimal extends to, use that power of 10 as the denominator, enter each number to the right of the decimal point as the numerator, and simplify. For example, looking at the number 0.1234, the number 4 is in the fourth decimal place which constitutes 104, or 10,000. This would make the fraction
 1234 10000
, which simplifies to
 617 5000
, since the greatest common factor between the numerator and denominator is 2. Similarly, fractions with denominators that are powers of 10 (or can be converted to powers of 10) can be translated to decimal form using the same principles. Take the fraction
 1 2
for example. To convert this fraction into a decimal, first convert it into the fraction
 5 10
. Knowing that the first decimal place represents 10-1,
 5 10
can be converted to 0.5. If the fraction were instead
 5 100
, the decimal would then be 0.05, and so on. Beyond this, converting fractions into decimals requires the operation of long division.

### Common Engineering Fraction to Decimal Conversions

In engineering, fractions are widely used to describe the size of components such as pipes and bolts. The most common fractional and decimal equivalents are listed below.

 64th 32nd 16th 8th 4th 2nd Decimal Decimal(inch to mm) 1/64 0.015625 0.396875 2/64 1/32 0.03125 0.79375 3/64 0.046875 1.190625 4/64 2/32 1/16 0.0625 1.5875 5/64 0.078125 1.984375 6/64 3/32 0.09375 2.38125 7/64 0.109375 2.778125 8/64 4/32 2/16 1/8 0.125 3.175 9/64 0.140625 3.571875 10/64 5/32 0.15625 3.96875 11/64 0.171875 4.365625 12/64 6/32 3/16 0.1875 4.7625 13/64 0.203125 5.159375 14/64 7/32 0.21875 5.55625 15/64 0.234375 5.953125 16/64 8/32 4/16 2/8 1/4 0.25 6.35 17/64 0.265625 6.746875 18/64 9/32 0.28125 7.14375 19/64 0.296875 7.540625 20/64 10/32 5/16 0.3125 7.9375 21/64 0.328125 8.334375 22/64 11/32 0.34375 8.73125 23/64 0.359375 9.128125 24/64 12/32 6/16 3/8 0.375 9.525 25/64 0.390625 9.921875 26/64 13/32 0.40625 10.31875 27/64 0.421875 10.715625 28/64 14/32 7/16 0.4375 11.1125 29/64 0.453125 11.509375 30/64 15/32 0.46875 11.90625 31/64 0.484375 12.303125 32/64 16/32 8/16 4/8 2/4 1/2 0.5 12.7 33/64 0.515625 13.096875 34/64 17/32 0.53125 13.49375 35/64 0.546875 13.890625 36/64 18/32 9/16 0.5625 14.2875 37/64 0.578125 14.684375 38/64 19/32 0.59375 15.08125 39/64 0.609375 15.478125 40/64 20/32 10/16 5/8 0.625 15.875 41/64 0.640625 16.271875 42/64 21/32 0.65625 16.66875 43/64 0.671875 17.065625 44/64 22/32 11/16 0.6875 17.4625 45/64 0.703125 17.859375 46/64 23/32 0.71875 18.25625 47/64 0.734375 18.653125 48/64 24/32 12/16 6/8 3/4 0.75 19.05 49/64 0.765625 19.446875 50/64 25/32 0.78125 19.84375 51/64 0.796875 20.240625 52/64 26/32 13/16 0.8125 20.6375 53/64 0.828125 21.034375 54/64 27/32 0.84375 21.43125 55/64 0.859375 21.828125 56/64 28/32 14/16 7/8 0.875 22.225 57/64 0.890625 22.621875 58/64 29/32 0.90625 23.01875 59/64 0.921875 23.415625 60/64 30/32 15/16 0.9375 23.8125 61/64 0.953125 24.209375 62/64 31/32 0.96875 24.60625 63/64 0.984375 25.003125 64/64 32/32 16/16 8/8 4/4 2/2 1 25.4