Unitary Method

In mathematics, the unitary method is a powerful and easy technique for solving problems: first find the value of one unit, then use it to determine the value of any number of units. This method works on the principle of proportionality: when one quantity changes, the other changes in direct or inverse proportion. 

The reason it’s called the unitary method is simple: everything starts with understanding one unit. Once we know that, we can solve a wide range of unitary method problems, including ratio and proportion, cost, time, distance, and more. 
 

For example, imagine a car travels 60 km in 2 hours. How far will it travel in 1 hour, and how far in 5 hours?
Solution: 
Distance in 2 hours = 60 km.
So, distance in 1 hour = \(60 ÷ 2 = 30\) km.
Then the distance travelled in 5 hours \(= 30 × 5 = 150\) km.

Use the following steps to apply the unitary method:

• Determine what the unit depicts in the given problem. For example, one meter of cloth or one liter of milk.

• To find out the value of one unit, we should divide the total value by the given quantity.

• Multiply the value of a single unit with the given number of units to get the final value.

For example: The cost of 10 meters of cloth = $150, calculate the cost of 5 meters of cloth.

Cost of 1 meter = \(\frac{150}{10} = $15\).

Therefore, the cost of 5 meters = \(15 × 5 = $75\).

The unitary method comes in two different variants. The two types are mentioned below:

Direct Variation: In direct variation, if one quantity increases or decreases, the other quantity also increases or decreases in the same proportion. For example, a worker paints 10 square meters of a wall in 1 hour. If the amount of time increases, the amount of work done also increases in the same proportion. 

That is, if the worker paints 10 m² in 1 hour, then in 3 hours, the work done will be calculated as:
Work in 1 hour = 10 m²
Work in 3 hours = \(10 × 3 = 30\) m²
 

Indirect Variation: A mathematical relationship where two quantities are indirectly connected to each other, means they are in indirect variation. In the indirect variation, when one quantity increases, the other decreases, and vice versa. For example, the time taken to walk a fixed distance varies inversely with the speed. If you walk faster, the time decreases and if you walk slower, the time increases. 

A student needs to walk a fixed distance of 3 km to reach school. When walking at the speed of 6 km/h,
the time taken \(= 3 ÷ 6 = 0.5\) hours (30 minutes).
When walking at a speed of 3 km/h, the time taken \(= 3 ÷ 3 = 1\) hour.

Here, as speed decreases, the time increases.
And as speed increases, the time decreases.
 

The unitary method helps us compare two quantities by first finding the value of one unit. When working with ratios and proportions, this method is very useful because it allows us to convert quantities to a standard unit before comparing them.

Let’s understand this through an example:
Emma earns $2,400 per month, while Sarah earns $33,600 per year. Both of them spend $1,800 per month on their expenses. Find the ratio of their monthly savings.

Solution: 
Monthly income of Emma \(= $2,400\).
Monthly expenditure \(= $1,800\).
Then, savings \(= 2,400 − 1,800 = $600\).

Annual income of Sarah \(= $33,600\).
Monthly income \(= 33,600 ÷ 12 = $2,800\).
Monthly expenditure \(= $1,800\).
Then, savings \(= 2,800 − 1,800 = $1,000\).

Ratio of their savings \(= 600 : 1000\).
By dividing both by 200: 
\(600 ÷ 200 = 3\)
\(1000 ÷ 200 = 5\)
Therefore, the ratio of Emma’s and Sarah’s savings \(= 3:5\).

Finding it difficult to solve problems using the unitary method? These below-mentioned tips and tricks should help you solve them easily:

• As a first step, you need to identify the context of the problem, i.e., what needs to be determined.

• Check if the problem involves direct or indirect variation.

• Write the quantities in a simple ratio form to make it easier to solve.

• Ensure that you follow all the steps and that the final result includes the proper unit.

• Always practice by applying it in real-life problems. For e.g., if 10 pens cost $100, what is the cost of 5 pens? To find the cost of one pen, we will divide 100 by 10. So \(\frac{100}{10} = 10\); therefore, 5 pens will cost \(10 × 5 = $50\).
  
   
• Encourage children to see the unitary method in everyday situations like shopping, cooking, travel distance, time taken for chores, etc. This helps them understand how to use the unitary method naturally.
  
   
• Students learn better with visual aids. Create simple tables, charts, or a unitary-method worksheet showing 1 unit, multiple units, and direct and indirect variation. 
  
   
• When students solve unitary method questions, remind them that the first step is always to find the value of 1 unit. Then, multiply or divide as needed. Help students stick to this sequence through repeated unitary method examples.
  
   
• Parents and teachers can prepare small practice tasks like: If three books cost $27, find the cost of 1, and then seven books. And, if a child walks more slowly, will the time increase or decrease? This will help students to strengthen their understanding of direct and indirect variations. 
  
   
• Show children how the unitary method connects to other topics such as ratios, proportions, percentages, speed–distance–time, and cost problems. This makes the process more flexible and easier to apply across topics.

The unitary method is a mathematical concept that has numerous applications in various fields. We will now learn how they can be applied:

• Calculating product prices and discounts: The unitary method helps students find the cost of a single item or a group of items and discounts. For example, if 5 notebooks cost $25, then the cost of 1 notebook is \($25 ÷ 5 = $5\). So, the cost of 8 notebooks is \(8 × $5 = $40\).

• Estimating travel time: Students can use the unitary method to figure out how long a trip will take based on how much distance is covered in one unit of time. For example, if a cyclist travels 6 km in 20 minutes, then:
  distance per minute \(= 6 ÷ 20 = 0.3\) km.
  Then, the distance covered in 1 hour (60 minutes): 
  \(0.3 × 60 = 18\) km.

• Finding monthly salary from daily wages: By knowing how much a worker earns in one day, their approximate monthly salary can be calculated. For example, if a gardener earns $50 per day, then in 30 days:
  Then, the monthly salary \(= 50 × 30 = $1500\).

• Estimating fuel requirements for a journey: The unitary method helps students understand how much fuel is needed based on fuel consumption per unit distance. 
  For example, if a car uses 1 liter of fuel to travel 15 km, then for a 90 km trip:
  Fuel needed \(= 90 ÷ 15 = 6\) liters.

• Planning meal quantities for families: Families can apply the unitary method to scale recipes according to the number of people. For example, if a recipe uses 200 g of rice to serve 2 people, then to serve 5 people:
  Rice needed per person \(= 200 ÷ 2 = 100\) g
  Total rice needed \(= 100 × 5 = 500\) g.