TIME & WORK Problems

---

TIME & WORK : Formula Sheet

Type - 1 : @. Two Persons Together

Shortcut Formula : If A completes a work in x days and B completes the same work in y days, then Together Time = xy / x+y​

i.  Problem: A = 8 days, B = 24 days, Together Time = ?

Together Time = 8 × 24 / 8+24​ = 192/32 = 6 days 

Ans: 6 days

ii. Problem: A = 6 days, B = 12 days, Together Time = ?

Together Time = 6 × 12 / 6 + 12​ = 72/18 = 4 days 

Ans: 4 days

==================================================

Tip

• For three persons: Together Time = xyz / xy + yz + zx​​


• For two persons: Together Time = xy / x + y​​


• More workers ⇒ More efficiency ⇒ Less time

===================================================

Type - 2 : @. Three Persons Together

Shortcut Formula

If:

• A completes a work in x days


• B completes the same work in y days


• C completes the same work in z days

Then,

Together Time = xyz / xy + yz + zx​

i. Problem: A = 4 days, B = 6 days, C = 12 days, Together Time = ?

Shortcut Method: Together Time = Together Time = xyz / xy + yz + zx = 4 × 6 × 12 / (4×6) + (6×12) + (12×4)​ = 288 / 24 + 72 + 48 = 288/144 $$=2\text{ days}$$

Ans: 2 days

LCM Method :                                                                                                                                                  Formula Sheet

LCM of 4, 6, and 12 = 12 units

Efficiencies

• A's efficiency : 12 ÷ 4 = 3


• B's efficiency : 12 ÷ 6 = 2


• C's efficiency : 12  ÷12 = 1

Total efficiency : 3 + 2 + 1 = 6

Time taken together: 12 / 6 = 2 days

Ans: A, B, and C together can complete the work in 2 days.

====================================================

Type - 3 : @. Efficiency Method:

Problem: A = 12 days, B = 18 days, Find the together time using the LCM Method.

1st: Find Total Work : LCM of 12 and 18 = 36

So, Total Work = 36 units

2nd: Find Efficiencies

• A's Efficiency: 36 ÷ 12 = 3, A's efficiency = 3 units/day


• B's Efficiency: 36 ÷ 18 = 2, B's efficiency = 2 units/day


• Total Efficiency: 3 + 2 = 5, Total efficiency = 5 units/day

3rd: Find Together Time

Time = Total Work / Total Efficienc​ = 36/5 = 7 1/5 days = $$7.2\text{ days}$$

Ans: A and B together can complete the work in 7 1/5 days (or 7.2 days).

Shortcut Formula : Together Time = 12×18 / 12+18 = 216/30​ = 36/5 = 7 1/5 days

Ans: 7.2 days.

==========================================================

Type - 4 : @. Fraction Work

Problem: A can do a fraction of work in some days. B can do another fraction of the same work in some days.

• First convert fractional work into full work.


• Then apply the together-time formula.

Main Concept: If a person completes 1/n work in d days, then: Full Work Time = d × n

Formula: Full Work Time = Given Days × Denominator, Always convert partial work into complete work first.

Ex: A can do 1/5 of a job in 10 days. B can do 1/3 of the same job in 25 days. In how many days can they complete the job together ?

1st: Convert A'S work

A does 1/5 work in 10 days.

So, A completes full work in: 10 × 5 = 50 days

2nd: Convert B'S work

B does 1/3 work in 25 days.

So, B completes full work in: 25 × 3 = 75 days

3rd: Apply "A" - "B" formula

Together Time = 50 × 75 / 50 + 75 = 3750 / 125 = $$30\text{ days}$$

Ans: A and B together can complete the job in 30 days.

Tip

• Partial work must be converted into full work first.


• After converting, use the normal A + B together formula.

Formula: Together Time = AB / A+B​, where A and B are the full-work times

Problem: A can do 1/4 work in 8 days., B can do 1/5 work in 10 days.

Solⁿ

A's full work time: 8 × 4 = 32 days

B's full work time: 10 × 5 = 50 days

Together time: 32 × 50 / 32 + 50 = 1600/82 = 19.51 days

Ans: A and B together can complete the work in approximately 19.5 days.

==========================================================

Type - 5 : Efficiecy Ratio

Problem: A and B together can complete a work in some days. Their efficiency ratio is given. Find the time taken by A and B separately.

Main Concept

• Total Work = Together Time × Total Efficiency


• Individual Time = Total Work ÷ Individual Efficiency

Important Rule: Efficiency and Time are opposite to each other.

• Higher Efficiency → Less Time


• Lower Efficiency → More Time

Ex: A and B together can complete a work in 12 days. Their efficiency ratio is 2 : 3. Find the time taken by A and B separately.

Solⁿ

1st: Add Efficiencies : - A : B = 2 : 3

Total Efficiency = 2 + 3 = 5 units/day

2nd: Find Total Work

Total Work = Together Time × Total Efficiency = 12 × 5 = 60 units

3rd: Find Individual Time

• A's Time = Total Work ÷ A's Efficiency = 60 ÷ 2 = 30 days


• B's Time = Total Work ÷ B's Efficiency = 60 ÷ 3 = 20 days

Ans: A alone can complete the work in 30 days., B alone can complete the work in 20 days.

Tip

• Higher efficiency means fewer days.


• Lower efficiency means more days.


• Never confuse Efficiency Ratio with Time Ratio.

Remember

If Efficiency Ratio = a : b, Then Time Ratio = b : a

Shortcut Formula

If, Together Time = T, Efficiency Ratio = a : b

Then, A's Time = T(a + b)/a, B's Time = T(a + b)/b

Problem: A and B together finish a work in 15 days. Their efficiency ratio is 3 : 2. Find A's and B's separate time.

Solⁿ

1st, Add Efficiencies : A : B = 3 : 2

Total Efficiency = 3 + 2 = 5 units/day

2nd : Find Total Work

Total Work = 15 × 5 = 75 units

3rd : Find Individual Time

A's Time = 75 ÷ 3 = 25 days

B's Time = 75 ÷ 2 = 37.5 days

Ans: A alone can complete the work in 25 days., B alone can complete the work in 37.5 days.

Shortcut: When Efficiency Ratio is 3 : 2, Time Ratio becomes 2 : 3, Together Time = 15 days

So,

• A's Time = 15 × (3 + 2)/3 = 25 days


• B's Time = 15 × (3 + 2)/2 = 37.5 days

Same Answer.