PROBLEM & SOLUTION
Example.1) If the wages of 15 workers for 5 days are $ 1500, find the wages of 25 workers for 4 days.
Ans.) As per the given condition –
Wages of 15 workers for 5 days is $ 1500
$ 1500
Wages of 1 worker for 5 days will be = ------------
15
[ as per the direct variation ]
$ 1500
Wages of 1 worker for 1 day will be = ------------- 15 X 5
[ as per the direct variation ]
$ 1500 X 25
Wages of 25 workers for 1 day will be = ---------------- 15 X 5
[ as per the direct variation ]
$ 1500 X 25 X 4
Wages of 25 workers for 1 day will be = --------------------- 15 X 5
[ as per the direct variation ] = $ 2000
So, the wages of 25 workers for 4 days will be $ 2000 (Ans.)
Example.2) In an army camp of 150 soldiers has enough ration to least 60 days. If 50 more soldiers join the camp after 10 days, how long will the remaining ration last ?
Ans.) at the end of 10 days the camp has enough ration to provide for 150 soldiers for 60 – 10 = 50 days. Since the number of soldiers increases, the ration will last for less/fewer days than planned. In other words, an increase in the number of soldiers will mean a decrease in the number of days. Hence, this is a case of Inverse proportion
1
Thus, the ratio of the number of soldiers = -----------------------
Ration of number of days
150 1
So, ---------- = ---------,
200 50
where x is the number of days for which the ration will start
150 x
So, ---------- = -----------
200 50
150 X 50
So, x = -------------- = 37.5 days
200
( 150 : 200 = 3 : 4 is the multiplying ratio )
Another way –
As per the given condition-
150 soldiers have a ration for 60 days,
But after 10 days 50 soldiers has added, so after 10 days remaining ration for 150 + 50 = 200 soldiers is for 60 – 10 = 50 days
So, now if 150 soldiers goes the said ration by 50 days
Then, only 1 soldier goes the said ration by ( 50 X 150 ) days. [ as per the inverse variation ]
( because 1 soldiers will take more time to finish the same food or rations than 150 soldiers takes to finish, so their inverse variation is applied ) 50 X 150
Now, 200 soldiers will take to finish the same ration = ------------- 200
[ as per the inverse variation ]
( we have placed the total number of soldiers in denominator place, because 200 soldiers will take to finish the ration or food less/fewer days than 1 soldier take to finish the same food ) = 37.5 days
So, 37.5 days long will the remaining ration last for 200 soldiers. (Ans.)
Example.3) If 8 men working 5 hours a day do a job in 15 days, in how many days will 4 men working 6 hours a day do the same job ?
Ans.) As per the given condition -
8 men working 5 hours a day do a job in 15 days
So, 1 man working 5 hours a day a job in 15 X 8 days
[ as per the inverse variation ]
( here, 1 man will take more days to execute the same work than 8 men works with 5 hours of working, so it will go as multiplication mode as per inverse variation )
So, 1 man working 1 hour a day a job in 15 X 8 X 5 days
[ as per the inverse variation ]
( here, 1 man with 1 hour working will take more days to execute the same work than 8 men works with 5 hours of working, so it will go as multiplication mode as per inverse variation )
15 X 8 X 5
Now, 4 men working with 1 hour a day a job in = ---------------
4
[ as per the inverse variation ]
( here, 4 men
with 1 hour working will take less/fewer days to execute same work than 1 man works
with 1 hours of working, so it will go as division mode as per inverse
variation )
Now, 4 men working with 6 hour a day a job in = ---------------
4 X 6
[ as per the inverse variation ]
( here, 4 men with 6 hours working will take less/fewer days to execute the same work than 1 man works with 1 hour of working, so it will go as division mode as per inverse variation ) = 25 days
So, we can conclude that, 4 men working 6 hours a day do the same job by 25 days. (Ans.)
Example.4) If 6 men or 9 women earn $ 900 in a day then how much will 8 men and 12 women earn in a day ?
Ans.) As per the given condition -
6 men’s earning in a day = $ 900
900
So, 1 man’s earning would be = ----------- 6
[ as per the direct variation ]
( 1 man’s earning would be less than 6 men’s earning, that’s why this is in division mode as per direct variation )
900 X 8
So, 8 men’s earning would be = ------------- 6
= $ 1200
( 8 man’s earning would be more than 1 man’s earning, that’s why this is in multiplication mode as per direct variation )
9 women’s earning in a day = $ 900 900
So, 1 woman’s earning would be = ----------- 9
[ as per the direct variation ]
( 1 woman’s earning would be less than 9 women’s earning, that’s why this is in division mode as per direct variation ) 900 X 12
So, 12 women’s earning would be = --------------- 9
[ as per the direct variation ]
= $ 1200
( 12 women’s earning would be more than 1 woman’s earning, that’s why this is in multiplication mode as per direct variation )
Hence, the total earnings of 8 men & 12 women is $ 1200 + $ 1200 = $ 2400 (Ans.)
Example.5) If in a hostel, there are 200 students who consume 250 kg of rice in 8 days. How many students will consume 300 kg of rice in 15 days ?
Ans.) As per the given condition –
If, 250 kg rice consume in 8 days by 200 students
200
1 kg rice consume in 8 days by = ------------ 250
[ as per the direct variation ]
( 1 kg rice in 8 days will be consumed by 200 students which will be less/fewer days than 250 kg rice in 8 days, so we will go in division mode as per direct variation ) 200 X 8
1 kg rice consume in 1 day by = -------------- 250
[ as per the inverse variation ]
( 1 kg rice in 1 day will be consumed by 200 students which will be more days than 1 kg rice in 8 days, so we will go in multiplication mode as per inverse variation )
200 X 8 X 300
300 kg rice consume in 1 days by = ------------------- 250 X 15
[ as per the direct variation ]
( 300 kg rice in 1 day will be consumed by number of students which will be more than the students who will consume 1 kg rice in 1 days, so we will go in multiplication mode as per direct variation )
200 X 300 X 8
300 kg rice consume in 15 days by = ------------------ 250 X 15
[ as per the inverse variation ]
( 300 kg rice in 15 days will be consumed by a number of students which will be less than the students who will consume 300 kg rice in 1 day, so we will go in division mode as per direct variation ) = 128 students
So, 128 students will consume 300 kg of rice in 15 days (Ans.)