 ONLINE MATHEMATICS NOTES

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Topic 4: Commercial Arithmetics

1. RATIOS AND PROPORTIONS

1.1. Ratios

A ratio is a type of fraction which makes it possible to compare two or more quantities of the same unit or kind. Suppose there are 20 cows and 40 bulls in the farm. We can express this as We can say that the ratio of the number of cows to the number of bulls is 1:2 (read as “1 to 2”). Note that ratio has no unit.

Example

A chicken laid 15 brown eggs and 10 white eggs. What is the ratio of white eggs to brown eggs?

Solution

The ratio is Example

Write the following ratios in their simplest form

a) 10 to 5                      b) 60:15                        c) 18:45

Solution

a) b) c) Sharing

Ratios are often used to describe sharing.

Example

Ndoli and Kabera share 20 sweets in the ratio 2:3. How many does each gets?

Solution

Total shares 2+3=5

Ndoli gets of 20 sweets, that is sweets

Kabera gets of 20 sweets, that is sweets.

1.2. Direct and inverse proportions

If an increase in one quantity, say x, causes an increase in another quantity, say y, in the same ratio or if a decrease in one quantity causes a decrease in another quantity; then the quantities are in direct proportion or are directly proportional. We write where k is the constant of proportionality.

In some cases an increase in one quantity leads to a decrease in the other. Such quantities are said to be inversely proportional. In this case we write where k is the constant of proportionality.

Example

Four people can do a job in nine days. How long will one person take to do the same job under the same conditions?

Solution

Let x be the number of people and y be the number of days. The two quantities are inversely proportional because if the number of people decreases, the number of days must be increased.

So, . But Thus, .

Now, if Therefore, one person takes 36 days.

Example

Copy and complete the following table, if the two quantities are in direct proportion.

 No. of shelves 1 5 10 ... Planks of wood ... 17.5 ... 52.5

Solution

Let x be the number of shelves and y be planks of wood. Since the two quantities are in direct proportion, we have . But If Complete table:

 No. of shelves 1 5 10 15 Planks of wood 3.5 17.5 35 52.5

Joint proportion

There are cases where more than two quantities vary. Consider a quantity x varies directly with y. The same quantity x varies inversely with z.

In this case we write: Example

If 5 tractors can clear a field of 100 acres in one week, how long will it take 3 tractors to clear a field of 180 acres under the same conditions?

Solution

Let x be the number of tractors, y be the size of the land and z be the time to be used (days). Then x is directly proportional to y and inversely proportional to z. That is But  Now,  Hence 3 tractors will use 21 days.

Alternative method Hence 3 tractors will use 21 days.

2. PERCENTAGES

2.1. Definition

A percent, which is a ratio of a number to 100, is also called a rate. Here, the word rate is treated as a comparison of a quantity to the whole. For example, 5% (read as 5 percent) is the ratio of 5 to 100. A percent can be expressed as a fraction or as a decimal: Example

Express as a percentage: Mike ran 10 km out of 50 km

Solution

10 km out of 50 km is So, Mike ran 20% of 50 km.

Percentage error

When we use a measuring device such as a ruler to obtain a measurement, the accuracy and precision of the measure is dependent on the type of instrument used and the care with which it is used.

Error is the absolute value of the difference between a value found experimentally and the true theoritical value and we find the percentage error by the following formula Example

The width and lenght of a rectangle are 13 cm and 84 cm espectively, the true lenght of the diagonal, found using the Pythagorean Theorem, is 85 cm. A student drew this rectangle and, using a ruler, found the measure of the diagonal to be cm. Find the percentage error of measurement.

Solution

The percentage error of measurement is 2.2. Multiplier

A multiplier is a factor that increase or decrease the proportion of a given quantity.

There are a decreasing multiplier and an increasing multiplier. Consider the price of the book being reduced by 15%, the percentage of the selling price is 100% - 15% = 85%.

But, . We say that 0.85 is the multiplier of the price of the book.

If is the initial amount, the final amount is An increasing multiplier

An increasing multiplier is a factor that increases the proportion of a given quantity.

Here, Example

Increase 200kg by 8%. What is the multiplier?

Solution

8% of 200kg is New quantity is 200kg + 16kg = 216kg

The multiplier is 100% + 8% =108%=1.08

Using multiplier we can also find the new quantity: Example

Increase 600 by 15%

Solution

The multiplier is 100% + 15% =115% = 1.15

The new value is A decreasing multiplier

A decreasing multiplier reduces the proportion of a given quantity.

Here, Example

Decrease 72kg by 40%

Solution

The mutiplier is 100% - 40% = 60% = 0.6

The new quantity is Reverse percentage

Reverse percentage calculations are problems in which we are given a % of an amount and we have to find the original amount. Let be the final amount and  be the orginal amount Where is the original amount and is the final amount.

Example

A laptop has been reduced in a sale by 20%. A sale price is 320,000 Frw. What was the original cost of the laptop?

Solution

The multiplier is 100% - 20%=80%=0.8, the final cost 320,000 Frw

The original cost is Thus, the original cost is 400,000 Frw

Example

The price of a house has increased by 30% since it was first built 23 years ago. The house now cost 260,000 Frw. What was the value of the house when it was first built?

Solution

The multiplier is 100% + 30%=130%=1.3, the final cost is 260,000 Frw

The original cost is The value of the house when it was first built is 200,000 Frw.

3. MIXTURES

3.1. Definition

In chemistry, a mixture is a material made up of two or more different substances which are mixed. A mixture refers to the physical combination of two or more substances in which the identities are retained and are mixed in the form of solutions, suspensions and colloids. e.g. air can be taken as a mixture of nitrogen and oxygen (but can be taken as a pure substance if composition does not change in the problem at hand).

Most substances found in nature are mixtures of pure chemical elements or compounds: air, natural gas, seawater (but also tap water), coffee, wine, gasoline, antifreeze, body fluids, etc.

Mixture problems are ones where two different solutions are mixed together resulting in a new final solution.

We will use the following table to help us solve mixture problems: The first column is for the amount of each item we have. The second column is labeled “part”. If we mix percentages we will put the rate (written as a decimal) in this column. If we mix prices we will put prices in this column. Then we can multiply the amount by the part to find the total. Then we can get an equation by adding the amount and/or total columns that will help us solve the problem and answer the questions. These problems can have either one or two variables.

Example

A chemist has 70 ml of a 50% methane solution. How much of a 80% solution must she add so the final solution is 60% methane?

Solution

Set up the mixture table. We start with 70, but don't know how much we add, that is x.The part is the percentages, 0.5 for start, 0.8 for add. Add amount column to get final amount.The part for this amount is 0.6 because we want the final solution to be 60% methane. Multiply amount by part to get total.  Thus, 35 ml must be added.

Example

A farmer has two types of milk, one that is 24% butterfat and another which is 18% butterfat. How much of each should he use to end up with 42 gallons of 20% butterfat?

Solution  Thus, 14 gal of 24% and 28 gal of 18% should be used.

4. APPLICATIONS