Question

Pooja and Ritu can do a piece of work in $ 17\dfrac{1}{7} $ days. If one day's work of Pooja be three fourth of one day's work of Ritu; find in how many days each will do the same work alone.

Answer 1

Hint:Time and work are directly proportional to each other.
If two or more people work together simultaneously, then their individual works for the same amount of time can be added to give the work done by them together in that same amount of time.

Use variables to express the works and compare the works for the same amount of time in all the cases to form some equations and solve them.

Complete step by step solution:
Let's say that the 1 day's work of Ritu is $ 4x $ units. Therefore, 1 day's work of Pooja will be $ 3x $ units.

Let us calculate the amount of work done by both of them individually in $
17\dfrac{1}{7}=\dfrac{120}{7} $ days.

Since time and work are directly proportional to each other, we can multiply both of them with the same quantity, without affecting the rates of the individuals.

Ritu: 1 day's work is $ 4x $ units.
⇒ $ 1\times \dfrac{120}{7} $ day's work is $ (4x)\times \dfrac{120}{7} $ units.
⇒ $ \dfrac{120}{7} $ day's work is $ (4x)\times \dfrac{120}{7} $ units.

Pooja: 1 day's work is $ 3x $ units.
⇒ $ 1\times \dfrac{120}{7} $ day's work is $ (3x)\times \dfrac{120}{7} $ units.
⇒ $ \dfrac{120}{7} $ day's work is $ (3x)\times \dfrac{120}{7} $ units.

When they work together simultaneously, the total amount of work done by them will be:
Ritu + Pooja: $ \dfrac{120}{7} $ day's work is $ (4x)\times \dfrac{120}{7}+(3x)\times \dfrac{120}{7} $ units.
⇒ $ \dfrac{120}{7} $ day's work is $ (7x)\times \dfrac{120}{7} $ units.

⇒ $ \dfrac{120}{7} $ day's work is $ 120x $ units.

It means that the total amount of the piece of work is $ 120x $ units.
Now,

Ritu: 1 day's work is $ 4x $ units.
⇒ $ 1\times 30 $ day's work is $ (4x)\times 30 $ units.
⇒ $ 30 $ day's work is $ 120x $ units.
And,

Pooja: 1 day's work is $ 3x $ units.
⇒ $ 1\times 40 $ day's work is $ (3x)\times 40 $ units.
⇒ $ 40 $ day's work is $ 120x $ units.

Therefore, Pooja alone will do the piece of work in 40 days and Ritu alone will do the piece of work in 30 days.

Note:If two or more people work together "alternately", then their times (not necessarily same) as well as their works in those amounts of time, both get added together.

In the case of working together simultaneously, the times (must be same) are NOT added
together. Ratio is derived from Latin, literally ‘reckoning’, from rat- ‘reckoned’, e.g. ratify; and it means the same as rate in mathematics.

The statements which involve words like per (per cent, per day, per person, per hour), each,
every, etc. are all based on rates. i.e. direct proportions.