Solution:
Total number of invalid votes = 15 % of 560000
= 15/100 × 560000
= 8400000/100
= 84000
Total number of valid votes 560000 – 84000 = 476000
Percentage of votes polled in favour of candidate A = 75 %
Therefore, the number of valid votes polled in favour of candidate A = 75 % of 476000
= 75/100 × 476000
= 35700000/100
= 357000
2. A shopkeeper bought 600 oranges and 400 bananas. He found 15% of oranges and 8% of bananas were rotten. Find the percentage of fruits in good condition.
Solution:
Total number of fruits shopkeeper bought = 600 + 400 = 1000
Number of rotten oranges = 15% of 600
= 15/100 × 600
= 9000/100
= 90
Number of rotten bananas = 8% of 400
= 8/100 × 400
= 3200/100
= 32
Therefore, total number of rotten fruits = 90 + 32 = 122
Therefore Number of fruits in good condition = 1000 - 122 = 878
Therefore Percentage of fruits in good condition = (878/1000 × 100)%
= (87800/1000)%
= 87.8%
3. Aaron had $ 2100 left after spending 30 % of the money he took for shopping. How much money did he take along with him?
Solution:
Let the money he took for shopping be m.
Money he spent = 30 % of m
= 30/100 × m
= 3/10 m
Money left with him = m – 3/10 m = (10m – 3m)/10 = 7m/10
But money left with him = $ 2100
Therefore 7m/10 = $ 2100
m = $ 2100× 10/7
m = $ 21000/7
m = $ 3000
We will follow the following steps for converting a percentage into a fraction:
Step I: Obtain the given percentage. Let it be x %.
Step II: Remove the percentage sign (%) and then divide the number by 100. Therefore, x % = x/100
Step III: Reduce the fraction obtained to its lowest terms as required.
1. Express each of the following percentage into fraction in lowest terms:
(i) 16 % = 16/100 = 4/25
(ii) 48 % = 48/100 = 12/25
(iii) 5 % = 5/100 = 1/20
(iv) 25 % = 25/100 = 1/4
(v) 115 % = 115/100 = 23/20
(vi) 1 % = 1/100
2. Convert 27 per cent to fraction.
27 % = 27/100
3. Convert each of the following percentage as fractions in the lowest form:
(i) 24 % = 24/100 = 6/25
(ii) 62 % = 62/100 = 31/50
(iii) 30 % = 30/100 = 3/10
(iv) 75 % = 75/100 = ¾
(v) 10 % = 10/100 = 1/10
4. Convert each of the decimal percentage as fractions in the lowest form:
(i) 3.5 % = 35/10 % = 35/10 × 1/100 = 7/100
(ii) 0.5 % = 5/10 % = 5/10 × 1/100 = 1/200
(iii) 30.2% = 322/10 % = 322/10 × 1/100 = 322/1000 = 151/500
(iv) 0.4 % = 4/10 % = 4/10 × 1/100 = 4/1000 = 1/250
(v) 0.375 % = 375/1000 % = 375/1000 × 1/100 = 375/100000 = 3/800
5. Convert each of the fraction percentage as fractions in the lowest form:
(i) 32/5 % = 17/5 % = 17/5 × 1/100 = 17/500
(ii) 2/5 % = 2/5 × 1/100 = 1/250
(iii) 162/3 % = 50/3 % = 50/3 × 1/100 = 50/300 = 1/6
(iv) 51/8 % = 41/8 % = 41/8 × 1/100 = 41/800
(ii) 2/5 % = 2/5 × 1/100 = 1/250
(iii) 162/3 % = 50/3 % = 50/3 × 1/100 = 50/300 = 1/6
(iv) 51/8 % = 41/8 % = 41/8 × 1/100 = 41/800
We will follow the following steps for converting a percentage into a ratio:
Step I: Obtain the percentage.
Step II: Convert the given percentage into fraction by dividing it by 100 and removing percentage symbol (%).
Step III: Reduce the fraction obtained in step II in the simplest form.
Step IV: Write the fraction obtained in step III as a ratio.
1. Express each of the following percentage as ratios in the simplest form:
(i) 46 % = 46/100 = 23/50 = 23 : 50
(ii) 20 % = 20/100 = 1/5 = 1 : 5
(iii) 125 % = 125/100 = 5/4 = 5 : 4
(iv) 34% = 34/100 = 17/50= 17 : 50
(v) 1 % = 1/100 = 1 : 100
2. Express each of the following fraction percentage into ratio in lowest term:
(i) 3/4 % = 3/4 × 1/100 = 3/400 = 3 : 400
(ii) 62/3 % = 20/3 % = 20/3 × 1/100 = 20/300 = 1 : 15
(iii) 3/5 % = 3/5 × 1/100 = 3/500 = 3 : 500
(iii) 62/5 % = 32/5 % = 32/5 × 1/100 = 32/500 = 8/125 = 8 : 125
(iv) 53/8 % = 43/8 % = 43/8 × 1/100 = 43/800 = 43 : 800
(ii) 62/3 % = 20/3 % = 20/3 × 1/100 = 20/300 = 1 : 15
(iii) 3/5 % = 3/5 × 1/100 = 3/500 = 3 : 500
(iii) 62/5 % = 32/5 % = 32/5 × 1/100 = 32/500 = 8/125 = 8 : 125
(iv) 53/8 % = 43/8 % = 43/8 × 1/100 = 43/800 = 43 : 800
3. Express each of the following decimal percentage as ratios in the simplest form:
(i) 16.5 % = 165/10 % = 165/10 × 1/100 = 165/1000 = 33/200 = 33 : 200
(ii) 0.4 % = 4/10 % = 4/10 × 1/100 = 4/1000 = 1/250 = 1 : 250
(iii) 2.5 % = 25/10 % = 25/10 × 1/100 = 25/1000 = 1/40 = 1 : 40
(iv) 10.10 % = 1010/100 % = 1010/100 × 1/100 = 1010/10000 = 101/1000 = 101 : 1000
We will follow the following steps for converting a ratio into a percentage:
Step I: Obtain the ratio. Let the ratio be x : y
Step II: Convert the given ratio into the fraction x/y.
Step III: Multiply the fraction obtained in step II by 100 and put the percentage sign(%).
1. Express each of the following ratio into percentage:
(i) 6 : 5 = 6/5 = (6/5 × 100) % = 600/5 % = 120 %
(ii) 8 : 25 = 8/25 = (8/25 × 100) % = 32 %
(iii) 10 : 50 = 10/50 = (1/5 × 100) % = 20 %
(iv) 4 : 5 = 4/5 = (4/5 × 100) % = 80 %
(v) 7 : 25 = 7/25 = (7/25 × 100) % = 28 %
2. Convert each of the following ratios as fraction percentage:
(i) 20 : 70 = 20/70 = (20/70 × 100) % = 2000/70 % = 200/7 %
(ii) 17 : 24 = 17/24 = (17/24 × 100) % = 1700/24 % = 425/6 %
(iii) 128 : 175 = 128/175 = (128/175 × 100) % = 512/7 %
(iv) 2 : 3 = 2/3 = (2/3 × 100) % = 200/3 %
(v) 1 : 15 = 1/15 = (1/15 × 100) % = 100/15 % = 20/3 %
3. Convert each of the following ratios into decimal percent:
(i) 1 : 8 = 1/8 = (1/8 × 100) % = 100/8 % = 12.5 %
(ii) 30 : 80 = 30/80 = (30/80 × 100) % = 3000/80 % = 75/2 % = 37.5 %
(iii) 5 : 8 = 5/8 = (5/8 × 100) % = 500/8 % = 125/2 % = 62.5 %
(iv) 11 : 16 = 11/16 = (11/16 × 100) % = 1100/16 % = 275/4 % = 68.75 %
(v) 8 : 125 = 8/125 = (8/125 × 100) % = 800/125 % = 32/5 % = 6.4 %
We will follow the following steps for converting a percentage into a decimal:
Step I: Obtain the percentage which is to be converted into decimal
Step II: Remove the percentage sign (%) and divide it by 100.
Step III: Express the fraction in the decimal form
Remember: Remove % sign and move the decimal two places to the left.
1. Express each of the following percentage as decimals:
(i) 23 % = 23/100 = 0.23
(ii) 16 % = 16/100 = 0.16
2. Express each of the following decimal percentage as decimals:
(i) 1.75 % = 1.75/100 = (1.75 × 100)/(100 × 100) = 175/10000 = 0.0175
(ii) 7.5 % = 7.5/100 = (7.5 × 10)/(100 × 10) = 75/1000 = 0.075
(iii) 31.5 % = 31.5/100 = (31.5 × 10)/(100 × 10) = 315/1000 = 0.315
(iv) 0.18 % = 0.18/100 = (0.18 × 100)/(100 × 100) = 18/10000 = 0.0018
(v) 231.2 % = 231.2/100 = (231.2 × 10)/(100 × 10) = 2312/1000 = 2.312
3. Express each of the following fraction percentage as decimals:
(i) 4/5 % = 0.8 % = 0.8/100 = (0.8 × 10)/(100 × 10) = 8/1000 = 0.008
(ii) 9/20 % = 0.45 % = 0.45/100 = (0.45 × 100)/(100 × 100) = 45/10000 = 0.0045
(iii) 1/80 % = 0.0125 % = 0.0125/100 = (0.0125 × 10000)/(100 × 10000) = 125/1000000 = 0.000125
(iv) 13/50 % = 0.26 % = 0.26/100 = (0.26 × 100)/(100 × 100) = 26/10000 = 0.0026
(v) 44/5 % = 8.8 % = 8.8/100 = (8.8 × 10)/(100 × 10) = 88/1000 = 0.088
We will follow the following steps for converting a decimal into a percentage:
Step I: Obtain the number in decimal form.
Step II: Multiply the number in decimal form by 100 and put percent sign (%)
Note: When we multiply the decimal number by 100, then we need to shift the decimal point two places to the right (add zeros if necessary).
I. Express each of the following decimal as percent:
(i) 0.8 = 0.8 × 100 % = 80 %
(ii) 1.2 = 1.2 × 100 % = 120 %
(iii) 7.1 = 7.1 × 100 % = 710 %
(iv) 10.1 = 10.1 × 100 % = 1010 %
(v) 31.3 = 31.3 × 100 % = 3130 %
(vi) 123.7 = 123.7 × 100 % = 12370 %
(vii) 101.9 = 101.9 × 100 % = 10190 %
II. Conversion of decimal into percent:
(i) 0.29 = 0.29 × 100 % = 29 %
(ii) 0.25 = 0.25 × 100 % = 25 %
(iii) 0.01 = 0.01 × 100 % = 1 %
(iv) 0.51 = 0.51 × 100 % = 51 %
(v) 13.01 = 13.01 × 100 % = 1301 %
(vi) 201.17 = 201.17 × 100 % = 20117 %
(vii) 100.11 = 100.11 × 100 % = 10011 %
III. Convert the following decimal into a percentage:
(i) 0.083 = 0.083 × 100 % = 8.3 %
(ii) 0.225 = 0.225 × 100 % = 22.5 %
(iii) 0.001 = 0.001 × 100 % = 0.1 %
(iv) 0.003 = 0.003 × 100% = 0.3 %
(v) 0.107 = 0.107 × 100 % = 10.7 %
(vi) 1.037 = 1.037 × 100 % = 103.7 %
(vii) 21.275 = 21.275 × 100 % = 2127.5 %
Note: Decimals shifts by 2 places to the right.
How to find the percentage of the given quantity?
We know, a percentage is a fraction with denominator as 100 i.e. % = 1/100. Therefore, to determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
We will follow the following steps for finding a percentage of a given number:
Step I: Obtain the number. Let the number be m.
Step II: Obtain the required percentage. Let it be R %.
Step III: To find R % of m, multiply m by R and then divide by 100; i.e. R % of m = R/100 × m
Following examples will help us to find the percentage of the given quantity using the above procedure.
1. Find 40 % of 240
Solution:
We know that R % of m is equal to R/100 × m.
So, we have 40 % of 240
40/100 × 240
= 96
2. 10 % of 1 hour
Solution:
We know that R % of m is equal to R/100 × m.
So, we have 10 % of 1 hour
10 % of 60 minutes (Since, 1 hour = 60 minutes)
= 10/100 × 60 minutes
= 6 minutes
3. Find 15 % of $250.
Solution:
We know that R % of m is equal to R/100 × m.
So, we have 15 % of $250.
15/100 × 250
= $75/2
= $37.5
4. Find 120 % of 25 km
Solution:
120 % of 25 km
= (120/100 × 25) km
= 30 km
5. Find 10 % of 400 kg
Solution:
10 % of 400 kg
= (10/100 × 400) kg
= 40 kg
6. Find the number whose 8 % is 72.
Solution:
Let the required number be m, then
8 % of m = 72
⇒ 8/100 × m = 72
⇒ m = 72 × 100/8
m = 900
Therefore, the required number is 900.
Word problems to find the percentage of the given quantity:
7. What is the sum of the money of which 15 % of $225?
Solution:
Let the required sum of money be $m.
15 % of $m = $225
⇒ 15/100 × m = 225
⇒ m = (225 × 100)/15
⇒ m = 1500
Therefore, sum of the money = $1500
8. In a public show 75 % of the seats were filled. If there were 600 seats in the hall, how many seats were vacant?
Solution:
First process:
75 % of 600
= 75/100 × 600
= 450
Therefore, the number of vacant seats = 600 - 450 = 150.
Second process:
Total percentage of seats = 100.
Percentage of filled seats = 75.
Therefore, percentage of vacant seats = 100 – 75 = 25.
25 % of 600
= 25/100 × 600
= 150.
Thus, the number of vacant seats is 150.
Note: Total is always 100%.
Finding how much percentage one quantity is of another. Suppose there are two quantities and we want to find what percent of one quantity is of the other quantity or find how many hundredths of one quantity should be taken so that it is equal to the second quantity.
We will follow the following steps for expressing one quantity as the percentage (percent) of another quantity.
Step I: Quantity to be compared is taken as a numerator.
Step II: Quantity with which it is to be compared is taken as a denominator.
Step III: Quantities are expressed in the fraction form then the fraction is multiplied by 100 %.
Note: The two quantities must be of the same kind and must have the same units.
Suppose, we have to express x as the percentage of y; then the formula is:
Percentage = x/y × 100
Note: Both x and y must have the same units.
We will apply the concept of solving some real-life problems by using the formula for finding how much percentage one quantity is of another.
Solve examples of what percent is one number of another number:
1. Sam scored 36 marks out of 60. Express the marks in percentage.
Solution:
Therefore, required percent = (36/60 × 100) %
= 60%
2. Express 80 ml as a percent of 400 ml
Solution:
Therefore, required percent = (80/400 × 100) %
= 20 %
3. Express 1 hour 36 minutes as the percent of 2 hours 40 minutes.
Solution:
We know, 1 hour = 60 minutes.
Therefore, 1 hour 36 minutes = (60 + 36) minutes = 96 minutes and
2 hours 40 minutes = (120 + 40) minutes = 160 minutes
Required percent = (96/160 × 100) %
= 60 %
4. Find the number if 12 % of it is 60.
Solution:
Let the number be m
Then 12 % of m = 60
⇒ 12/100 × m = 60
⇒ m = (60 × 100)/12
Therefore, the required number = 500
Word problems for finding how much percentage one quantity is of another:
1. What percent of $15 is 75 cents?
Solution:
We know, $1 = 100 cents
$15 = (15 × 100) cents = 1500 cents
Therefore, required percent = (75/1500 × 100) %
= 5 %
2. What percent of 70 kg is 2.1 kg?
Solution:
Required percent = (2.1 kg/70 kg × 100) %
= 210/70 %
= 3 %
Therefore, 2.1 kg is 3 % of 70 kg.
3. 296 is what per cent of 3700?
Solution:
Let m % of 3700 = 296
m/100 × 3700 = 296
m = (296 × 100)/3700
m = 29600/3700
m = 8
Therefore, 8 % of 3700 is 296.
How to find percentage of a number?
We will follow the following steps for finding percent of a number:
Step I: Obtain the given number
Step II: Multiply the number by the required percentage i.e.,
x % of a = x/100 × a
Solved examples to find the percent of a number:
1. Find 17 % of $ 1700
Solution:
17 % of $ 1700
= 17/100 × 1700
= $289
2. Find 10 % of 900
Solution:
10 % of 900
= 10/100 × 900
= 90
3. Find 25/8 % of 160.
Solution:
25/8 % of 160
= (25/8)/100 × 160
= 25/800 × 160
= 4000/800
= 5
4. Find the number if 35 % of it is 280 km.
Solution:
Let the required number be m.
Then 35 % of m = 280
⇒ 35/100 × m = 280
⇒ m = 280 × 100/35
⇒ m = 28000/35
⇒ m = 800
Therefore, 35 % of 800 km is 280 km
How to find a new number when a number is increase or decrease by a given percent?
(i) If a number is increased by x %, then the new number = (1 + x/100) × given number.
(ii) If a number is decreased by x %, then the new number = (1 – x/100) × given number.
Solved examples:
1. What is the new number if increase the number 300 by 20 %.
Solution:
New number = (1 + 20/100) × 300
= (100 + 20)/100 × 300
= 120/100 × 300
= 120 × 3
= 360
Therefore, the new number = 360
2. What is the new number if decreased the number 360 by 30 %.
Solution:
New number = (1 – 30/100) × 360
= (100 – 30)/100 × 360
= 70/100 × 360
= 25200/100
= 252
Therefore, the new number =252
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