In the previous section we saw how percentage is used for comparison of quantities. We also saw that the sum of the percentages in a problem is always 100. In this section, we continue our discussion:
• Mr.A saves 30% of his monthly income
• 25% of the annual budget of the municipality is reserved for the health care of the people
• Mr.B has decided to give 3% of the apples that he harvested to his friend.
The above statements give us an approximate idea about the 'size of the quantity'. On many occasions, this approximate idea will be sufficient. But if we want to do any calculations, this will not be sufficient. For example, It is said that Mr. A saves 30% of his income. Will this be sufficient to obtain a loan from the bank, to buy a car? To confirm that, we need to know the exact amount that Mr.A saves every month. If we know the monthly income, we can easily calculate the 30% of it. Let us see how it is done:
Let the monthly income be Rs.15000. We want to know how much is 30% of 15000.
• 30% means 30 per 100. Which is same as 30 for every 100
• So for every ₹100, he saves ₹30.
♦ If there are 2 'hundreds' in his income, he will save 2 × 30 = ₹ 60
♦ If there are 3 'hundreds' in his income, he will save 3 × 30 = ₹ 90
♦ If there are 4 'hundreds' in his income, he will save 4 × 30 = ₹ 120
And so on. So we need to know how many 'hundreds' are there in his income.
Dividing 15000 by 100 will directly give us the 'number of hundreds' which are present in 15000. So we can write:
• Number of hundreds present in 15000 = 15000⁄100 =150
• So there are 150 'hundreds' in 15000.
• For each of these 'hundreds', he saves ₹30. So the total amount saved = 150 × 30 = ₹4500
Thus we got 30% of 15000. All that we did was this:
We divided 15000 by 100. And then this result was multiplied with 30. That is: (15000⁄100 ) × 30.
This can be rearranged as: 15000 × ( 3⁄100 ).
■ So we multiplied the monthly income by the 'percentage written in fraction form'.
■ An even simpler method is to multiply with the decimal instead of the fraction. This will give the same result:
15000 × 0.3 = 4500 (∵ 30⁄100 =0.3)
Let us now take the second example. We want to know the exact amount that is reserved for health care. In other words, we want to how much is 25% of the annual budget.
• Let the annual budget be ₹6500000/-
• How many hundreds are there in 6500000?
• We get the answer by simple division: 6500000/100 = 65000
• So the number of hundreds = 65000.
•For each of these hundreds, ₹25 is kept aside for health care.
• So the total amount kept aside = 65000 × 25 = 1625000
• That is., 25% of 6500000 = 1625000
As in the previous example, we will get this same result in just one step:
■ 6500000 × 25⁄100 = 1625000.
■ This is same as 6500000 × 0.25 = 1625000. (since 25⁄100 = 0.25)
Third example:
Let the number of apples that was harvested be 2350
Number of hundreds = 2350/100 = 23.50
Here we do not get the 'number of hundreds' as a whole number. But it does not affect the calculations. It only means that there are 23 'full hundreds' and 0.5 (ie., half) of a 'full hundred'.
For each of the 'full hundreds' Mr.B will give 3 apples. For the one 'half hundred', he will give 1.5 apples
So number of apples given to friend = (23 × 3) + (0.5 × 3) = 69 + 1.5 =70.5
[This is same as 23.5 × 3 =70.5]
That is., 3% of 2350 = 70.5
As in the previous examples, we will get the same result in just one step:
■ 2350 × 3⁄100 = 70.5
■ This is same as 2350 × 0.03 = 70.5 (∵ 3⁄100 = 0.03)
70.5 will be rounded off to 71, and so 71 apples will be given.
We will now see some solved examples:
Solved example 8.12
A student wants to buy a book which costs ₹325. His friend agreed to give 30% of the cost. How much money did the friend give?
Solution:
Cost of book = 325
Donation from friend = 30% of 325 = 325 × 0.3 = ₹97.5
Solved example 8.13
A tank contains 750 litres of water. 15% of this is to be used for washing clothes. How many litres of water will be used for washing clothes?
Solution:
• Total quantity of water = 750 litres
• Quantity for washing clothes = 15% of 750 = 750 × 0.15 =112.5 litres.
Solved example 8.14
An alloy has 20% zinc, 35% alluminium, and the rest copper. Find the quantity of each of these metals in a sample of the alloy weighing 1.2kg
Solution:
• % of zinc = 20
• % of alluminium = 35
• Let the % of copper be x
• Sum of the above = 20 +35 +x . This must be equal to 100%
• So we can write 55 +x =100
∴ x = 45%
• Total weight of the sample = 1.2kg = 1200 grams
• Weight of zinc = 20% of 1200 = 1200 × 0.2 =240 grams =0.24 kg
• Weight of alluminium = 35% of 1200 = 1200 × 0.35 =420 grams = 0.42 kg
• Weight of copper = 45% of 1200 = 1200 × 0.45 =540 grams = 0.54 kg
In the next section we will see more solved examples.
Percentage of a quantity
In day to day life we hear many statements like:• Mr.A saves 30% of his monthly income
• 25% of the annual budget of the municipality is reserved for the health care of the people
• Mr.B has decided to give 3% of the apples that he harvested to his friend.
The above statements give us an approximate idea about the 'size of the quantity'. On many occasions, this approximate idea will be sufficient. But if we want to do any calculations, this will not be sufficient. For example, It is said that Mr. A saves 30% of his income. Will this be sufficient to obtain a loan from the bank, to buy a car? To confirm that, we need to know the exact amount that Mr.A saves every month. If we know the monthly income, we can easily calculate the 30% of it. Let us see how it is done:
Let the monthly income be Rs.15000. We want to know how much is 30% of 15000.
• 30% means 30 per 100. Which is same as 30 for every 100
• So for every ₹100, he saves ₹30.
♦ If there are 2 'hundreds' in his income, he will save 2 × 30 = ₹ 60
♦ If there are 3 'hundreds' in his income, he will save 3 × 30 = ₹ 90
♦ If there are 4 'hundreds' in his income, he will save 4 × 30 = ₹ 120
And so on. So we need to know how many 'hundreds' are there in his income.
Dividing 15000 by 100 will directly give us the 'number of hundreds' which are present in 15000. So we can write:
• Number of hundreds present in 15000 = 15000⁄100 =150
• So there are 150 'hundreds' in 15000.
• For each of these 'hundreds', he saves ₹30. So the total amount saved = 150 × 30 = ₹4500
Thus we got 30% of 15000. All that we did was this:
We divided 15000 by 100. And then this result was multiplied with 30. That is: (15000⁄100 ) × 30.
This can be rearranged as: 15000 × ( 3⁄100 ).
■ So we multiplied the monthly income by the 'percentage written in fraction form'.
■ An even simpler method is to multiply with the decimal instead of the fraction. This will give the same result:
15000 × 0.3 = 4500 (∵ 30⁄100 =0.3)
Let us now take the second example. We want to know the exact amount that is reserved for health care. In other words, we want to how much is 25% of the annual budget.
• Let the annual budget be ₹6500000/-
• How many hundreds are there in 6500000?
• We get the answer by simple division: 6500000/100 = 65000
• So the number of hundreds = 65000.
•For each of these hundreds, ₹25 is kept aside for health care.
• So the total amount kept aside = 65000 × 25 = 1625000
• That is., 25% of 6500000 = 1625000
As in the previous example, we will get this same result in just one step:
■ 6500000 × 25⁄100 = 1625000.
■ This is same as 6500000 × 0.25 = 1625000. (since 25⁄100 = 0.25)
Third example:
Let the number of apples that was harvested be 2350
Number of hundreds = 2350/100 = 23.50
Here we do not get the 'number of hundreds' as a whole number. But it does not affect the calculations. It only means that there are 23 'full hundreds' and 0.5 (ie., half) of a 'full hundred'.
For each of the 'full hundreds' Mr.B will give 3 apples. For the one 'half hundred', he will give 1.5 apples
So number of apples given to friend = (23 × 3) + (0.5 × 3) = 69 + 1.5 =70.5
[This is same as 23.5 × 3 =70.5]
That is., 3% of 2350 = 70.5
As in the previous examples, we will get the same result in just one step:
■ 2350 × 3⁄100 = 70.5
■ This is same as 2350 × 0.03 = 70.5 (∵ 3⁄100 = 0.03)
70.5 will be rounded off to 71, and so 71 apples will be given.
We will now see some solved examples:
Solved example 8.12
A student wants to buy a book which costs ₹325. His friend agreed to give 30% of the cost. How much money did the friend give?
Solution:
Cost of book = 325
Donation from friend = 30% of 325 = 325 × 0.3 = ₹97.5
Solved example 8.13
A tank contains 750 litres of water. 15% of this is to be used for washing clothes. How many litres of water will be used for washing clothes?
Solution:
• Total quantity of water = 750 litres
• Quantity for washing clothes = 15% of 750 = 750 × 0.15 =112.5 litres.
Solved example 8.14
An alloy has 20% zinc, 35% alluminium, and the rest copper. Find the quantity of each of these metals in a sample of the alloy weighing 1.2kg
Solution:
• % of zinc = 20
• % of alluminium = 35
• Let the % of copper be x
• Sum of the above = 20 +35 +x . This must be equal to 100%
• So we can write 55 +x =100
∴ x = 45%
• Total weight of the sample = 1.2kg = 1200 grams
• Weight of zinc = 20% of 1200 = 1200 × 0.2 =240 grams =0.24 kg
• Weight of alluminium = 35% of 1200 = 1200 × 0.35 =420 grams = 0.42 kg
• Weight of copper = 45% of 1200 = 1200 × 0.45 =540 grams = 0.54 kg
In the next section we will see more solved examples.
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