Direct and Inverse Proportions
Direct Proportion
Two quantities x and y are
said to be in direct proportion if an increase in one quantity results an
increase in the other quantity and a decrease in one quantity results a
decrease in the other quantity. If two quantities vary always in the same ratio,
then they are in direct proportion.
Examples for Direct Proportion
Principal and interest are
in direct proportion, because if the principal is more, the interest earned
will also be more.
The distance and time are
in direct proportion, because more the distance travelled, more will be the time taken (if speed remains the same).
Number of articles
purchased and the amount spent are in direct proportion, because purchase of
more articles will cost more money. If two quantities x and y vary directly in
such a way that x/y remains a constant, then this constant is called the
constant of direct proportion. If x α y, then x = k.y, where k is proportionality
constant x/y = k.
So, x1/y1 =
k and x2/y2 = k
Then, x1/y1 =
x2/y2.
Example 1: Rajat takes 3 hours to cover 150 km. Find the
distance he will travel in 8 hours.
Solution: Let distance covered = y. When time increases
the distance also increases. Therefore, they are in direct proportion, 3 : 8 = 150
: y → y = (150 × 8)/3 = 400 km. Rajat will travel 400 km in 8 hours.
Example 2: The cost price of 15 articles is Rs 4500. Find
the number of articles purchased for Rs. 1500.
Solution: Let articles purchased = x. When amount spent
decreases, then number of articles also decreases. So, they are in direct proportion
→ 15 : x = 4500 : 1500 → x =
(15 × 1500) / 4500 = 5
Example 3: The cost of 9 kg of sugar is Rs 360. Find the
cost of 18.5 kg of sugar.
Solution: Let the cost is Rs. x. When quantity increases, the cost
also increases. So, they are in direct proportion → 9 : 18.5 = 360 : x → x = Rs
740
Inverse Proportion
If two quantities x and y are such that an increase or decrease in one quantity leads to a corresponding decrease or increase in other quantity in the same ratio, then we can say they vary indirectly. Suppose 6 men can do a piece of work in 18 days, then 12 men can do the same job in 9 days. That means if we double the number of men, then number of days get halved. It means there is inverse relation between number of men and number of days.
In general, when two variables x and y are such that xy = k, where k is a non-zero constant, we say that y varies inversely with x. Inverse proportion is written as y α 1/x → y = k/x, where k is constant of proportionality. Thus, x1y1 = k and x2y2 = k. So, x1y1 = x2y2.
Examples for Inverse Proportion
Population and quantity of
food are in inverse proportion, because if the population increases, the food
availability decreases.
Speed and time are in inverse
proportion, because higher the speed, the lower is the time taken to cover a
distance.
Work and time are in inverse
proportion, because more the number of the workers, lesser will be the time
required to complete a job.
Example 4: 12 men can do a piece of work in 16 days. Find
the time taken to complete the same work by 8 men.
Solution: Let the time taken is x days. Less the number of men,
more will be the time taken to complete the work. Thus, the two quantities are
in inverse proportion.
12 × 16 = 8 × x
→ x = 24 days