In the previous section we saw the details about increase in percentage. In this section, we will see Decrease in percentage. This is just the opposite of increase, that we saw in the previous section. The equations can be derived in the same way. However we will do the steps again.
In day to day life we see statements such as:
• The price of some medicines decreased by 12%.
This same statement can be written in another form:
• There is a 12% decrease in the price of some medicines.
Both means the same.
• The price of Bakery products will decrease by 7%.
This same statement can be written in another form:
• There will be a 7% decrease in the price of Bakery products.
Both means the same.
• The number of fever patients decreased by 5%.
This same statement can be written in another form:
• There is a 5% decrease in the number of fever patients.
Both means the same.
When we read such statements on Newspaper, or hear them on TV, we will want to know the 'situation after the decrease'. For example, we will want to know the new prices after the decrease. Or we will want to know the number of fever patients after the decrease. Let us try to find the new values:
In the above statements, decrease is given as per cent. That is., per 100. Same as: 'for every 100'.
So, in the first example, for every hundred, there is a decrease of 12. That is every 100 in the original price becomes 100 - 12 = 88.
• Let the original price of such a medicine be ₹400/-
• How many hundreds are there in 400? It is 400⁄100 = 4
• So there are 4 hundreds in the original price of ₹400.
• Each of these hundreds become 88.
• So the new price is 4 × 88 = ₹352
The above calculations can be written in one step:
• Divide the original price by 100. Multiply the result by 'the new value of 100'. [The new value of 100 is 88]
• So we can write the calculation in just one line: 400⁄100 × 88
■ This can be rearranged as:
Second example:
• For every 100 there is a decrease of 7. That is., every 100 in the original price becomes 100 -7 = 93.
• Let the original price of a bakery product be ₹40 per kilogram.
• How many hundreds are there in 40? It is 40⁄100 = 0.40
• So there are 0.40 hundreds in 40
• Each of these hundreds become 93
• So the new price is 0.4 × 93 = ₹37.2
The above calculations can be written in one step:
• Divide the original price by 100. Multiply the result by 'the new value of 100'. [The new value of 100 is 93]
• So we can write the calculation in just one line: 40⁄100 × 93
■ This can be rearranged as:
Third example:
• For every 100 there is a decrease of 5. That is., every 100 in the original number becomes 100 - 5 = 95.
• Let the original number be 87.
• How many hundreds are there in 87? It is 87⁄100 = 0.87
• So there are 0.87 hundreds in 87.
• Each of these hundreds become 95.
• So the new number is 0.87 × 95 = 82.65
The above calculations can be written in one step:
• Divide the original number by 100. Multiply the result by 'the new value of 100'. [The new value of 100 is 95]
• So we can write the calculation in just one line: 87⁄100 × 95
■ This can be rearranged as:
From the above 3 examples we get a general method. It is the same Eq.8.1 that we wrote in the previous section. But the 'New value of 100' is different. So we will write it as:
Eq.8.3:
Where 'New value of 100' = 100 - 'decrease in percent' with out the % sign
We will now see some solved examples:
Solved example 8.27
It was heard on TV that, the price of computers will decrease by 3%. If the present price of a computer is ₹27500, what will be the new price?
Solution:
We have
• Given Original value = ₹27500
• Decrease in percent = 3
• ∴ New value of 100 = 100 - 3 = 97
• Substituting these in the equation, we get: New value = 27500 × 97⁄100 = 27500 × 0.97 = 26675
• So the new price = ₹26675
Solved example 8.28
A library has 15250 books. The owner of the library wants to decrease the number of books by 15%. How many books does he have to give away?
Solution:
We have
• Given Original value = 15250
• Decrease in percent = 15
• ∴ New value of 100 = 100 -15 = 85
• Substituting these in the equation, we get: New value = 15250 × 85⁄100 = 15250 × 0.85 = 12962.5
• So the new total number of books = 12962.5
∴ the number of books that he have to give away = 15250 - 12962.5 = 2287.5
• So the owner has to give away 2288 books
Solved example 8.29
The value of a machine decreases by 4% every year. If the present value is 1,20,000, then (i) what will be the value next year? (ii) What was the value the previous year?
Solution:
(a) We have
• Given Original value = 1,20,000
• Decrease in percent = 4
• ∴ New value of 100 = 100 - 4 = 96
• Substituting these in the equation, we get: New value = 120000 × 96⁄100 = 120000 × 0.96 = 115200
• So the value next year = 115200
(b) Here we can use the same Eq.8.3. But we have to find the original value
• Let original value = x
• New value = 1,20,000
• Decrease in percent = 4
• ∴ New value of 100 = 100 - 4 =96
• Substituting these in the equation, we get: 120000 = x × 96⁄100 ⇒ 120000 × 100 = 96x
∴ x = 12000000⁄96 = 125000
• So the value in the previous year = 125000
So we have seen how to obtain the new value when the percentage decrease is given. We must be able to do the reverse also. That is., when the original value, as well as the new value are given, we must be able to calculate the percentage decrease. This can be analysed as follows:
• Let the percentage decrease be x%.- - - (2)
• Then, every 100 in the original value will become (100 - x). That is., the 'new value of 100' = (100 - x)
• How many hundreds are there in the original value? It is equal to: original value⁄100
• When we multiply the 'number of hundreds' with the 'new value of 100', we get the 'new value'
• That is., we have to multiply (original value⁄100) by (100 - x) to get the 'new value'. So we can write:
In the last step we obtain 'x'. But 'x' is the percentage decrease as mentioned in (2) above. So we can write the equation:
Eq.8.4:
We must remember that, what we get from Eq.8.4 is the percentage decrease. So, when we get the result from Eq.8.4, we must immediately put a ‘%’ sign after it.
It may be noted that Eq.8.4 is very similar to Eq.8.2 that we derived in the previous section. The only difference is that, 'original value' and 'New value' has changed positions in the numerator.
We will now see some solved examples based on the above discussion:
Solved example 8.30
The price of Rice this year is ₹38. Previous year, it was ₹42. What is the percentage decrease in the price?
Solution:
We have:
• New value = ₹38
• Original value = ₹42
• Substituting these values in the equation, we get:
• Percentage decrease = (42 - 38)⁄42 × 100 = 4⁄42 × 100 = 9.52%
Solved example 8.31
The price of sugar increases by 15%. A house wife decides to reduce the consumption of sugar so that total expenditure will not increase. By what percentage should the consumption be decreased?
Solution:
Let the original price of sugar be ₹ x per kg
We have Eq.8.1:
• New value of 100 = 115
• So we get New value = New price of sugar per kg = x × 115/100 = 1.15x
• Let the original consumption be y1 kg and the new consumption be y2 kg
• Then original expense for sugar = y1 × x
• New expense for sugar = y2 × 1.15x
• The new consumption y2 must be less than the old consumption y1. By making such a reduction, the new expense on sugar will remain same as the old expense even if the price goes up. This can be explained based on the fig. below:
• When the two sides are equal we can write:
y1 × x = y2 × 1.15x ⇒ y1 = 1.15y2
• Now we can calculate the percentage decrease in consumption:
We have Eq.8.4 (derived above in this section):
• Here, original value = y1 = 1.15y2
• New value = y2
• So we can write:
Thus we get 13.04%
So we have completed the discussion on increase and decrease in percentage. In the next section we will see Properties of Triangles.
In day to day life we see statements such as:
• The price of some medicines decreased by 12%.
This same statement can be written in another form:
• There is a 12% decrease in the price of some medicines.
Both means the same.
• The price of Bakery products will decrease by 7%.
This same statement can be written in another form:
• There will be a 7% decrease in the price of Bakery products.
Both means the same.
• The number of fever patients decreased by 5%.
This same statement can be written in another form:
• There is a 5% decrease in the number of fever patients.
Both means the same.
When we read such statements on Newspaper, or hear them on TV, we will want to know the 'situation after the decrease'. For example, we will want to know the new prices after the decrease. Or we will want to know the number of fever patients after the decrease. Let us try to find the new values:
In the above statements, decrease is given as per cent. That is., per 100. Same as: 'for every 100'.
So, in the first example, for every hundred, there is a decrease of 12. That is every 100 in the original price becomes 100 - 12 = 88.
• Let the original price of such a medicine be ₹400/-
• How many hundreds are there in 400? It is 400⁄100 = 4
• So there are 4 hundreds in the original price of ₹400.
• Each of these hundreds become 88.
• So the new price is 4 × 88 = ₹352
The above calculations can be written in one step:
• Divide the original price by 100. Multiply the result by 'the new value of 100'. [The new value of 100 is 88]
• So we can write the calculation in just one line: 400⁄100 × 88
■ This can be rearranged as:
• For every 100 there is a decrease of 7. That is., every 100 in the original price becomes 100 -7 = 93.
• Let the original price of a bakery product be ₹40 per kilogram.
• How many hundreds are there in 40? It is 40⁄100 = 0.40
• So there are 0.40 hundreds in 40
• Each of these hundreds become 93
• So the new price is 0.4 × 93 = ₹37.2
The above calculations can be written in one step:
• Divide the original price by 100. Multiply the result by 'the new value of 100'. [The new value of 100 is 93]
• So we can write the calculation in just one line: 40⁄100 × 93
■ This can be rearranged as:
• For every 100 there is a decrease of 5. That is., every 100 in the original number becomes 100 - 5 = 95.
• Let the original number be 87.
• How many hundreds are there in 87? It is 87⁄100 = 0.87
• So there are 0.87 hundreds in 87.
• Each of these hundreds become 95.
• So the new number is 0.87 × 95 = 82.65
The above calculations can be written in one step:
• Divide the original number by 100. Multiply the result by 'the new value of 100'. [The new value of 100 is 95]
• So we can write the calculation in just one line: 87⁄100 × 95
■ This can be rearranged as:
Eq.8.3:
Where 'New value of 100' = 100 - 'decrease in percent' with out the % sign
We will now see some solved examples:
Solved example 8.27
It was heard on TV that, the price of computers will decrease by 3%. If the present price of a computer is ₹27500, what will be the new price?
Solution:
We have
• Given Original value = ₹27500
• Decrease in percent = 3
• ∴ New value of 100 = 100 - 3 = 97
• Substituting these in the equation, we get: New value = 27500 × 97⁄100 = 27500 × 0.97 = 26675
• So the new price = ₹26675
Solved example 8.28
A library has 15250 books. The owner of the library wants to decrease the number of books by 15%. How many books does he have to give away?
Solution:
We have
• Given Original value = 15250
• Decrease in percent = 15
• ∴ New value of 100 = 100 -15 = 85
• Substituting these in the equation, we get: New value = 15250 × 85⁄100 = 15250 × 0.85 = 12962.5
• So the new total number of books = 12962.5
∴ the number of books that he have to give away = 15250 - 12962.5 = 2287.5
• So the owner has to give away 2288 books
Solved example 8.29
The value of a machine decreases by 4% every year. If the present value is 1,20,000, then (i) what will be the value next year? (ii) What was the value the previous year?
Solution:
(a) We have
• Given Original value = 1,20,000
• Decrease in percent = 4
• ∴ New value of 100 = 100 - 4 = 96
• Substituting these in the equation, we get: New value = 120000 × 96⁄100 = 120000 × 0.96 = 115200
• So the value next year = 115200
(b) Here we can use the same Eq.8.3. But we have to find the original value
• Let original value = x
• New value = 1,20,000
• Decrease in percent = 4
• ∴ New value of 100 = 100 - 4 =96
• Substituting these in the equation, we get: 120000 = x × 96⁄100 ⇒ 120000 × 100 = 96x
∴ x = 12000000⁄96 = 125000
• So the value in the previous year = 125000
So we have seen how to obtain the new value when the percentage decrease is given. We must be able to do the reverse also. That is., when the original value, as well as the new value are given, we must be able to calculate the percentage decrease. This can be analysed as follows:
• Let the percentage decrease be x%.- - - (2)
• Then, every 100 in the original value will become (100 - x). That is., the 'new value of 100' = (100 - x)
• How many hundreds are there in the original value? It is equal to: original value⁄100
• When we multiply the 'number of hundreds' with the 'new value of 100', we get the 'new value'
• That is., we have to multiply (original value⁄100) by (100 - x) to get the 'new value'. So we can write:
Eq.8.4:
We must remember that, what we get from Eq.8.4 is the percentage decrease. So, when we get the result from Eq.8.4, we must immediately put a ‘%’ sign after it.
It may be noted that Eq.8.4 is very similar to Eq.8.2 that we derived in the previous section. The only difference is that, 'original value' and 'New value' has changed positions in the numerator.
We will now see some solved examples based on the above discussion:
Solved example 8.30
The price of Rice this year is ₹38. Previous year, it was ₹42. What is the percentage decrease in the price?
Solution:
We have:
• Original value = ₹42
• Substituting these values in the equation, we get:
• Percentage decrease = (42 - 38)⁄42 × 100 = 4⁄42 × 100 = 9.52%
Solved example 8.31
The price of sugar increases by 15%. A house wife decides to reduce the consumption of sugar so that total expenditure will not increase. By what percentage should the consumption be decreased?
Solution:
Let the original price of sugar be ₹ x per kg
We have Eq.8.1:
• So we get New value = New price of sugar per kg = x × 115/100 = 1.15x
• Let the original consumption be y1 kg and the new consumption be y2 kg
• Then original expense for sugar = y1 × x
• New expense for sugar = y2 × 1.15x
• The new consumption y2 must be less than the old consumption y1. By making such a reduction, the new expense on sugar will remain same as the old expense even if the price goes up. This can be explained based on the fig. below:
• When the two sides are equal we can write:
y1 × x = y2 × 1.15x ⇒ y1 = 1.15y2
• Now we can calculate the percentage decrease in consumption:
We have Eq.8.4 (derived above in this section):
• Here, original value = y1 = 1.15y2
• New value = y2
• So we can write:
Thus we get 13.04%
So we have completed the discussion on increase and decrease in percentage. In the next section we will see Properties of Triangles.
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