Unit Rate Formula: Unitary Method and Examples

The unit rate formula is something that we use in our daily routine without even recognizing it. Comparing quantities and calculating quantities that are equivalent to a change in one parameter with respect to another, like speed (distance covered in a period), are done by using the unit rate formula. Distance per unit time, interest rates, cost per kg or quantity per cost (1kg milk for INR 20), literacy rate, and some other statistics are examples where this formula is frequently used. The word ‘per’ has significance as it tells us that amount changes by a change in other parameters. It can be replaced by the ‘/’ symbol.

Read Also: Some Interesting Patterns

Key Terms: Ratio, Rate, Unitary method, Denominator, Speed, Division, Fraction, and Quantity.

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Ratio

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When comparing two quantities in our daily lives, we often calculate the difference between the two, that is, we subtract one from the other. 

• For example, let’s say Tina has 40 flowers and Giya has 35, then 40 – 35 = 5 which means that Tina has five more flowers than Giya. 
• This method is highly useful but fails when we need to compare very high quantities with a low value, for instance, the grasshopper's length with the ant’s length because the grasshopper's length is approximately 5 cm while that of ant is in millimeters. 
• So, in that case, we can calculate how many ants can be placed in a line so that the length of the line becomes equal to that of grasshoppers. 
• Another example, let us take the cost of a car INR 2,50,000 and cost of the bike be INR 50,000. The difference between the two is INR 2 lakh rupees while division gives us: 250000/50000 = 5.
• This means that a car costs five times the cost of a bike. The division gives us more clarity about the cost of a car concerning that of the bike.
• The above-given examples explain to us the role of division in expressing one quantity in terms of another, that is, how many times we need to add that quantity to make it equal to the other or by what number we need to multiply it to make it equal to the other. 
• This division or fraction is called Ratio. The ratio is a fraction of two quantities having the same units. Colon (‘:’) symbol is used to represent a ratio.

Check Important Notes for Unit Circle Formula

Consider the given example:

In a musical concert, there are 40 bodyguards and 8 singers & dancers. 

The Ratio of number of singers & dancers to the number of bodyguards = 40/8 = 5:1

The ratio of number of bodyguards to the total people (bodyguards + singers & dancers) = 40/48 = 5:6

• The ratio of two values can only be calculated if they are in the same unit. Like, a unit of length, for finding the ratio of heights of two persons, should be the same. Otherwise, the calculated answer will be wrong. 
• Also, in the above example, the ratio of the number of boys to the number of girls (4:3) is different than the ratio of the number of girls to the number of boys (3:4).

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Rate

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The ratio of two quantities that are related by some means so that they form a third quantity on division is called rate, also called rate of change which means how one quantity (in the numerator) changes with respect to the other quantity (in the denominator). 

• If the unit or quantity in which something is changing is absent, then the rate is generally expressed in terms of time.
• The rate of change can also be expressed in terms of other quantities such as cost of wheat per kg (weight), voltage per meter or pixels per meter (length), mass, or another quantity. 
• Speed, pulse, and the number of rotations have distance per unit time, heartbeats per unit time, and how many times a body rotates per unit time as their units respectively. 
• These are the examples where time is (unity) in the denominator. Literacy rate and cost have the number of literate people to the total number of people and price of the commodity per unit quantity (kg, or liter) as its units, respectively, and have no time in the denominator of the ratio. 

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Unit rate formula (or Unitary method)

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Unit rate formula is a formula or method through which we can calculate the value of a quantity taking other quantities as unity. We know the meaning of ratio. But ratio can have a denominator, not unity but while using the unit rate formula, the answer is always with the denominator as one. For calculating the rate of change of x with respect to y, the unit rate formula is given by:

Unit Rate = Ratio of two different quantities (with different units) such that they form a different quantity on division.

= x : y = x/y

This can be understood with the help of situations and examples.

Consider the situations given below:

• Let’s say one goes to the vegetable market to buy carrots. The cost of 2 kg of carrots is INR 24. The cost of 1 kg of carrots can be calculated using this formula.
• A scooter can travel a distance of 80 km before utilizing 4 liters of petrol. To calculate how much it can travel using 1 liter of petrol, we need a unit rate formula.
• The calculation of speed, that is, km per hour or meters per minute indicates that we need to apply the unitary method as the independent quantity, which is time, in this case, is taken as unity (per minute).

If we need the value of one unit first to find the value of the needed number of units, then we use a method called Unitary Method.

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Examples

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Various examples will be discussed below. 

Example 1

A bike covers 20 km in two hours. Calculate the distance it will cover in 6 hours?

Distance covered in 2 hours = 20 km
Distance covered in 1 hour = 20/2 = 10 km
Therefore, the distance covered in 6 hours = 10 * 6 = 60 km
Thus, distance covered in 6 hours will be 60 km.
Here one can see, to calculate the quantity asked, one has to find the rate first by dividing one quantity with another quantity of different units. Rate here is speed which is in units of km and hour, so, therefore, speed equals 10 km per hour. Hence, the bike covers 60 km in 6 hours.

Example 2

If the cost of 200 envelopes is INR 640, what will be the cost of 100 envelopes?

Cost of 200 envelopes =INR 640
Cost of one envelope = INR 640/200
Cost of 100 envelopes = (640/200) * 100 = INR 320

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Things to Remember

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• In many cases, comparison between quantities by using division makes more sense, as it means how many times one quantity is to the other quantity. This method is known as comparison by ratio.
• Before taking a ratio, checking the units is a must. If the quantities have different then convert them so that they have the same unit and then take the ratio.
• The same ratio can occur in different cases
• The order in which quantities are ratio is taken is important as 2:3 is different from 3:2.
• The method in which we first find the value of one unit and then the value of the required number of units is known as the unitary method or unit rate formula.

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