Ratio and Proportion : formulas, tricks | proportion and ratio examples
Ratio and Proportion :
Ratio:
In a cricket match, Sachin scored 160 runs and Rahul scored 80 runs. We can compare their scores in two ways :
1) By Subtraction :
Sachin's score - Rahul's score = 160 - 80
= 80
Thus, we see that Sachin has scored 80 runs more than Rahul.
2 ) By division :
Let us see how many times Sachin's score is that of Rahul's.
To do this, we shall divide Sachin's score by Rahul's.
Sachin's score / Rahul's score = 160 / 80 = 2 / 1.
Thus, we see, that Sachin's score is 2 times Rahul's score.
Definition of Ratio :
"When two quantities are compared by division, then the quotient of that division is called a 'ratio'. The colon ':' is used to express a ratio."
The ratio x/y is written as 'x:y' and is read as 'x is to y'.
In the ratio x : y , we call x as the first term or antecedent and y, the second term or consequent.
For example :
The ratio of 3 to 5 is written as 3 / 5 or ' 3 : 5 ' and is read as ' 3 is to 5 '.
Comparing two numbers by division means finding the ratio of the two numbers.
Some important formulae :
1) Mean proportional :
Mean proportional between x and y is √xy.
2) Third proportional :
Third proportional between m and n is ( n × n ) / m.
3) Fourth proportional :
Fourth proportional between p, q and r is qr / p.
Word Problems :
1) There are 50 boys and 40 girls in a class. Find the ratio of the number of boys to number of girls.
Ans. :
Ratio of the number of boys to the number of girls = Number of boys / Number of girls = 50 / 40 = 5 / 4.
Generally any ratio is written in the simplest from.
2) Write the ratio of Rs 4 to 40 paise.
Ans. :
Here, Rs 4 and 40 paise are amounts of money. That is, these quantities are of the same kind. But, their units are different.
Let us express the two quantities in the same unit.
Rs 4 = 400 paise.
Thus, the ratio of 400 paise to 40 paise
= 400 : 40
=10 : 1
Note :
When finding the ratio of two quantities of the same kind, we have to first express them in the same units. However, the ratio does not have a unit.
Proportion :
Let us consider the two ratio 6 : 8 and 9 : 12.
6 : 8 = 6 / 8 = 3 / 4 = 3 : 4
9 : 12 = 9 / 12 = 3 / 4 = 3 : 4
Here, the simplest forms of the two ratios 6 : 8 and 9 : 12 are equal.
When two ratios are equal, then the numbers in those ratios are said to be in proportion.
6 : 8 = 9 : 12 means that the four numbers 6, 8, 9 and 12 are in proportion.
When four numbers p, q, r, s are such that
p : q = r : s, then p, q, r, s are said to be in proportion.
Here p and s are called extremes while q and r are called mean terms.
Some examples :
Question: Determine if the numbers in each of these groups are in proportion.
1) 4, 12, 8, 24
Ans. :
4 : 12 = 4 / 12 = 1 / 3 = 1 : 3
8 : 24 = 8 / 24 = 1 / 3 = 1 : 3
Therefore, 4 : 12 = 8 : 24
The numbers 4, 12, 8, 24 are in proportion.
2) 4, 8, 3, 15
Ans. :
4 : 8 = 4 / 8 = 1 / 2 = 1 : 2
3 : 15 = 3 / 15 = 1 / 5 = 1 : 5
Therefore, the ratios 4 : 8 and 3 : 15 are not equal.
The numbers 4, 8, 3, 15 are not in proportion.
3) If 4 : 5 = x : 10 then x = ?
Ans. :
4 : 5 = x : 10
Therefore, 4 / 5 = x / 10
5x = 4 × 10
x = 4 × 10 / 5
x = 8.
4) If 12 bananas cost Rs 18, what will 4 bananas cost ?
Ans. :
Ratio of number of bananas = ratio of their costs.
Therefore, 12 / 4 = 18 / x
3 / 1 = 18 / x ------ reduced to the simplest form.
Now, 18 = 3 × 6 ( 6 times the numerator )
Therefore, number in the x is 1 × 6 = 6 ( 6 times the denominator )
Therefore, cost of 4 bananas Rs 6.
Or
12 / 4 = 18 / x
12 × x = 18 × 4
x = 18 × 4 / 12
x = 6.
Therefore, cost of 4 bananas Rs 6.
5) 10 pens cost Rs. 50. Hence find the cost of 25 such pens.
Ans. :
The ratio of the number of pens = the ratio of their costs.
Therefore, 10 / 25 = 50 / x
10 × x = 50 × 25
x = 50 × 25 / 10
x = 125.
In a cricket match, Sachin scored 160 runs and Rahul scored 80 runs. We can compare their scores in two ways :
1) By Subtraction :
Sachin's score - Rahul's score = 160 - 80
= 80
Thus, we see that Sachin has scored 80 runs more than Rahul.
2 ) By division :
Let us see how many times Sachin's score is that of Rahul's.
To do this, we shall divide Sachin's score by Rahul's.
Sachin's score / Rahul's score = 160 / 80 = 2 / 1.
Thus, we see, that Sachin's score is 2 times Rahul's score.
Definition of Ratio :
"When two quantities are compared by division, then the quotient of that division is called a 'ratio'. The colon ':' is used to express a ratio."
The ratio x/y is written as 'x:y' and is read as 'x is to y'.
In the ratio x : y , we call x as the first term or antecedent and y, the second term or consequent.
For example :
The ratio of 3 to 5 is written as 3 / 5 or ' 3 : 5 ' and is read as ' 3 is to 5 '.
Comparing two numbers by division means finding the ratio of the two numbers.
Some important formulae :
1) Mean proportional :
Mean proportional between x and y is √xy.
2) Third proportional :
Third proportional between m and n is ( n × n ) / m.
3) Fourth proportional :
Fourth proportional between p, q and r is qr / p.
Word Problems :
1) There are 50 boys and 40 girls in a class. Find the ratio of the number of boys to number of girls.
Ans. :
Ratio of the number of boys to the number of girls = Number of boys / Number of girls = 50 / 40 = 5 / 4.
Generally any ratio is written in the simplest from.
2) Write the ratio of Rs 4 to 40 paise.
Ans. :
Here, Rs 4 and 40 paise are amounts of money. That is, these quantities are of the same kind. But, their units are different.
Let us express the two quantities in the same unit.
Rs 4 = 400 paise.
Thus, the ratio of 400 paise to 40 paise
= 400 : 40
=10 : 1
Note :
When finding the ratio of two quantities of the same kind, we have to first express them in the same units. However, the ratio does not have a unit.
Proportion :
Let us consider the two ratio 6 : 8 and 9 : 12.
6 : 8 = 6 / 8 = 3 / 4 = 3 : 4
9 : 12 = 9 / 12 = 3 / 4 = 3 : 4
Here, the simplest forms of the two ratios 6 : 8 and 9 : 12 are equal.
When two ratios are equal, then the numbers in those ratios are said to be in proportion.
6 : 8 = 9 : 12 means that the four numbers 6, 8, 9 and 12 are in proportion.
When four numbers p, q, r, s are such that
p : q = r : s, then p, q, r, s are said to be in proportion.
Here p and s are called extremes while q and r are called mean terms.
Some examples :
Question: Determine if the numbers in each of these groups are in proportion.
1) 4, 12, 8, 24
Ans. :
4 : 12 = 4 / 12 = 1 / 3 = 1 : 3
8 : 24 = 8 / 24 = 1 / 3 = 1 : 3
Therefore, 4 : 12 = 8 : 24
The numbers 4, 12, 8, 24 are in proportion.
2) 4, 8, 3, 15
Ans. :
4 : 8 = 4 / 8 = 1 / 2 = 1 : 2
3 : 15 = 3 / 15 = 1 / 5 = 1 : 5
Therefore, the ratios 4 : 8 and 3 : 15 are not equal.
The numbers 4, 8, 3, 15 are not in proportion.
3) If 4 : 5 = x : 10 then x = ?
Ans. :
4 : 5 = x : 10
Therefore, 4 / 5 = x / 10
5x = 4 × 10
x = 4 × 10 / 5
x = 8.
4) If 12 bananas cost Rs 18, what will 4 bananas cost ?
Ans. :
Ratio of number of bananas = ratio of their costs.
Therefore, 12 / 4 = 18 / x
3 / 1 = 18 / x ------ reduced to the simplest form.
Now, 18 = 3 × 6 ( 6 times the numerator )
Therefore, number in the x is 1 × 6 = 6 ( 6 times the denominator )
Therefore, cost of 4 bananas Rs 6.
Or
12 / 4 = 18 / x
12 × x = 18 × 4
x = 18 × 4 / 12
x = 6.
Therefore, cost of 4 bananas Rs 6.
5) 10 pens cost Rs. 50. Hence find the cost of 25 such pens.
Ans. :
The ratio of the number of pens = the ratio of their costs.
Therefore, 10 / 25 = 50 / x
10 × x = 50 × 25
x = 50 × 25 / 10
x = 125.
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