Ratio and Proportion
I) Ratio :
The ratio of two quantities in the same units is the fraction that one quantity is of the other .
thus , the ratio a to b is the fraction a/b, written as a : b.
where a = antecedent
b = consequent
Example :
In the ratio 2 : 3, represents 2/3 with antecedent = 2 , consequent = 3
Rule :
The value of a ratio remain unchanged , if each one of its terms is multiplied or divided by a same non-zero number .
1) If a > b, then a : b is called a ratio of greater inequality .
2) If a < b, then a : b is called a ratio of less inequality.
3) If a > b and same positive number is added to each term of a : b , then the ratio is diminished .
4) If a < b and same positive number is added to each term of a : b , then the ratio is increased .
5) a2 : b2 is called duplicate ratio of a : b .
6) a3 : b3 is called triplet ratio of a : b .
7) √a : √b is called sub - duplicate ratio of a : b
8) ∛a : ∛b is called sub - triplicate ratio of a : b .
9) If a : b and c : d are two ratios , then ac : bd is called the ratio compounded of the given ratio .
II Proportion :
The equality of two ratios is called proportion .
If a : b = c : d , we say that a, b, c, d are proportional and , we write,
a : b : : c : d
where a & d = extremes term
b & c = means terms
We always have :
Product of Means = Product of Extremes
1) Mean proportional between a and b is √ab
2) If a : b : : c : d , we say that d is the fourth proportional to a, b, c.
3) If x is the third proportional to a , b then a : b : : b : x.
4) If a/b = c/d then a+b/a-b = c+d/c-d and
a-b/a+b = c-d/c+d
III Variation :
Two types of variation
I) Direct
In general word Directly Proportion mean the quantity x and y are increase or decrease simultaneously.
We say that x is directly proportional to y, If x = ky for some constant k and we write ,
x ∝ y
Example:
1) If the number of articles purchased increases, the total cost also increase.
2) More the money deposited in a bank, more is the interest earned.
II) Inverse
In general word Inversely Proportion mean the quantity x increase then y decrease or x decrease then y increase simultaneously.
Example :
1) As speed of a vehicle increase, the time taken to cover the same distance decreases.
2) For a given jobs, more the number of workers, less will be the time taken to complete the work.
Also , we say that x is inversely proportional to y , if
x = k/y for some constant k and we write ,
x ∝ 1/y .
Example :
1) An electric pole, 14 metres high, casts a shadow of 10 metres. Find the height of a tree that casts a shadow of 15 metres under similar conditions.
Solution :
Let the height of the tree be x metres.
|
Height of object |
14 |
x |
|
length of shadow |
10 |
15 |
This is a case of direct proportion
14 = x
10 15
14×15 = x
10
14 × 3 = x
2
21 = x
Thus , height of the tree is 21 metres .
2) If the weight of 12 sheets of thick paper is 40 grams, how many sheets of the same paper would weight 2500 grams.
Solution :
Let the number of sheets be x .
|
No. of sheets |
12 |
x
|
|
Weight |
40 |
2500 |
This is a case of direct proportion.
12 = x
40 2500
12 × 2500 = x
40
x = 750
Thus, the required no. of sheets of paper = 750 .
3) A train is moving at a uniform speed of 75 km/h
i) How far will it travel in 20 minutes ?
ii) Find the time required to cover a distance travelled ( in km ) in 20 minutes be x and time take ( in minutes ) to cover 250 km be y.
Solution :
Let the distance travelled ( in km ) in 20 minutes be x and time take ( in minutes ) to cover 250 km be y.
|
|
|
|
|
|
|
|
|
y |
This is a case of direct proportion.
i) 75 = x
60 20
75 × 20 = x
60
x = 25
So, the train will cover a distance of 25 km in 20 minutes.
ii) 75 = 250
60 y
y = 250 × 60
75
y = 200 minutes or 3 hours 20 minutes.
Therefore, 3 hours 20 minutes will be required to cover a distance of 250 kms.
4) The scale of a map is given as 1 : 30000000. Two cities are 4 cm apart on map. Find the actual distance between them.
Solution:
Let the map distance be x cm and actual distance be y cm, then
1 30000000 = x : y
1 = x
3 × 107 y
Since x = 4
So, 1 = 4
3 × 107 y
y = 4× 3× 107
y = 12× 107cm
y = 1200 km
Thus, two cities which are 4 cm apart on the map, are actually 1200 km away from each other.
5) 6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if any 5 pipes of the same type are used ?
Solution :
Let the desired time to fill the tank be x minutes.
|
Number of pipes |
|
|
|
Time ( in minutes ) |
|
|
80 × 6 = x × 5
80 × 6 = x
5
x = 96
Thus, time taken to fill the tank by 5 pipes is 96 minutes or 1 hour 36 minutes.
6) There are 100 students in a hostel. Food provision for them is for 20 days. How long will these provisions last, if 25 more students join the group ?
Solution :
Let the provisions last for x days.
|
Number of student |
|
|
| Number of days |
|
|
So,
100 × 20 = 125 x
100 × 20 = x
125
16 = x
Thus 16days provisions last if 25 more students join the group.
1) Divide Rs. 672 in the ratio 5 : 3 .
Sol.
Sum of the ratio terms = ( 5 + 3 ) = 8
5
First part = Rs 672 X ---- = Rs 420
8
3
Second part = Rs 672 X ----- = Rs 252
8
2) A mixture contains alcohol and water in the ratio 4 : 3 . If 5 liters of water is added to the mixture , the ratio becomes 4 : 5 . Find the quantity of alcohol in the gives mixture .
Sol.
Let quantity of alcohol 4x litres
and quantity of water 3x litres
Then
4x 4
---------------- = ------
3x + 5 5
20x = 4 ( 3x + 5 )
20x = 12x + 20
20x-12x = 20
8x = 20
20
x = ------
8
x = 2.5
quantity of alcohol = ( 4 X 2.5 )
= 10 litres .
Exercise :
1) If 0.75 : x : : 5 : 2 , then x is equal to :
a) 1.12 b) 1.20 c) 1.25 d) 1.30
2) If x : y = 5 : 2 , then ( 8x + 9y ) : ( 8x + 2y ) is :
a) 22 : 29 b) 26 : 61 c) 29 : 22 d) 61 : 26
3) Salaries of Ravi and Sumit are in the ratio 2 : 3 . If the salary of each is increased by Rs 4000 , the new ratio becomes 40 : 57 . What is Sumit's present salary ?
a) Rs. 17,000 b) Rs 20,000 c) Rs 25,000 d) None of these .
4) If 40% of a number is equal to two - third of another number , what is the ratio of first number to the second number ?
a) 2: 5 b) 3: 7 c) 5 : 3 d) 7 : 3
5) If 10% of x = 20 % of , then x : y is equal to
a) 1: 2 b) 2 : 1 c) 5 : 1 d) 10 : 1
Answer :
1) b 2) c 3) d 4) d 5) b
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