## What is Unitary Method?

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.

Let us look at an example. Assume the price of 10 dolls is $45. Now, if we want to find the price of 14 dolls from the given statement how will we do that?

First, we will find the price for 1 unit; that is, the price of one doll. Then, we will multiply that price with the required number of dolls; in this case, it is 14.

## How does Unitary Method Work?

This is an arithmetic method in which we find the value of a single unit from the value of the given units. Once we have the value of a single unit, we can then find the value of the required units by multiplying the single value unit with the required number of units or if we have the value of a certain number of units, we may find the required number of units after dividing the value given by the value of a single unit.

## Real-Life Examples

We have seen in the above example in the introduction part how we would come across the unitary method.

Let us see more on using the unitary method with some real-life examples. At the grocery stores, we see billboards that say 10 oranges for $6 or $10 for a dozen apples.

But do we always buy the exact 1 or 2 dozen apples or do we always buy the exact 10 oranges? No, we don’t.

Sometimes we may buy 3 dozen apples for the pie or we may even buy 8 oranges for the marmalade. How is the exact price calculated in that case?

That’s where the **unitary** method comes into play.

To know the exact price of 8 oranges we need to find the price of one orange and then we can multiply that price with 8 to know the amount we need to buy 8 oranges.

**Practice Problem**

Alex goes to buy colored balls and finds that a packet of 20 such balls costs $28.

Alex doesn’t have so much money. He has only $7 with him. How many balls can he buy for $7? Alex tries to find the price of one ball.

He divides 28 into 20 and finds 28 20 =1.4.

Now, he wants to find the number of units which can be bought for $7. So, he divides 7 by 1.4

and finds 7 1.4 =5.

Alex used the **unitary** method to find out that he can buy 5 balls for $7.

While solving the unitary method we must recognize which values are known and which value we must find.

If we consider Alex’s situation, the value of 20 balls was known, the amount of money he had with him was known.

Then what was unknown?

At first, the price of a single ball was unknown, and the number of balls Alex can afford with his money was also unknown.

If the situation was framed as: “If Alex wants to buy 5 balls, then how much money he has to pay?”, then the number of balls Alex wants to buy is known and the money he has to spend is unknown.

## Types of Variations

In solving by the unitary method, we need to understand and use two types of variations—direct and indirect variation.

### Direct Variation

A direct variation is a variation in which there is a direct connection or relationship between the entities. In other words, if one entity increases, the second one also increases, and vice-versa.

For example, the relation between the total cost and the number of products is a direct variation; if the number of products increases the total cost will also increase.

### Inverse Variation

An inverse variation is a variation in which there is an inverse connection or relationship between the entities. In other words, if one entity increases, the other one decreases, and vice-versa.

For example, the relation between the speed of a vehicle and the time taken to reach the destination is an inverse variation; if the speed of the vehicle increases, the time taken to reach the destination will decrease.

## Where is the Unitary Method Used?

Unitary methods are generally used in most arithmetic problems including:

- Ratio & Proportion
- Time & Work
- Percentages
- Speed, Distance & Time
- Commodity & Price etc.

Let us see one example on each topic.

### Ratio & Proportion

Nick earned $56 after selling 8 pairs of shoes and Allen earned $84 after selling 14 pairs of shoes. What is the ratio of their respective earning?

Nick's earnings after selling one pair of shoes are $ 56 8 = $7.

Allen's earnings after selling one pair of shoes are $ 84 14 = $6.

So, the ratio of their respective earnings is 7 : 6.

### Time & Work

If 10 men can plough a field in 6 days, in how many days will 12 men plough the field?

10 men can plough a field in 6 days.

1 man can plough the field in 10 × 6 = 60 days,

12 men can plough the field in 60 12 = 5 days.

### Percentages

If 60% of Rocky’s monthly salary is $900 then what is his annual salary?

When we work with percentages, we take 100% as one unit.

So, first, we will find 100% of Rocky’s salary that is $ 900 60 × 100 = $1,500.

Rocky’s one-month salary is $1,500.

Rocky’s annual salary is $1,500 × 12 = $18,000.

### Speed, Distance & Time

Lucy traveled a distance of 500 kilometers by motorcycle in 20 hours. How much time will she take to cover 750 kilometers?

Lucy traveled 500 kilometers in 20 hours,

In 1 hour, she travels 500 20 = 25 kilometers.

She will travel 750 kilometers in 750 25 = 30 hours.

So, Lucy will take 30 hours to cover 750 kilometers.

### Commodity & Price

If the price of 10 dolls is $ 45, then what should be the price of 14 dolls?

The price of 10 dolls is $ 45,

The price of 1 doll is $ 45 10 = $4.5.

So, the price of 14 dolls is $4.5 × 14 = $63.

## Formula

If the value of “n” products is “p”, then the value of a unit product “u” is p n .

The value of “m” such products is u × m = p n × m.

## Context and Applications

We have already seen in various examples how the unitary method helps in our daily lives.

- When we go out to buy groceries and essentials, it helps to find the correct price of products.
- When we travel it helps us to calculate the estimated time to reach if we are already aware of the moving speed.
- Finding estimated salary.
- Finding profit and loss in business.
- Finding estimated time needed to finish a work.

## Common Mistakes

Some of the common errors that may occur when dealing with the unitary method are:

- Failing to understand the question properly and failing to identify correct known and unknown values in the question.
- Failing to identify the variation – if it is direct or indirect variation.
- Making errors in calculations. Do not forget to cross-check calculations.

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