Samacheer Kalvi Class 8 Maths Solutions Chapter 4 Life Mathematics Exercise 4.5

 

Question 1. A fruit vendor bought some mangoes of which 10% were rotten. He sold \( 33\frac{1}{3}\% \) of the rest. Find the total number of mangoes bought by him initially, if he still has 240 mangoes with him.
Answer: Let 'x' be the total number of mangoes the fruit seller bought at first.
Number of rotten mangoes \( = \frac{10}{100} \times x = \frac{10}{100}x \)
Number of good mangoes \( = x - \frac{10}{100}x = \frac{100x - 10x}{100} = \frac{90}{100}x \) (1)
The vendor sold \( 33\frac{1}{3}\% \) of the good mangoes. This is equal to \( \frac{100}{3}\% \).
Mangoes sold \( = \frac{100}{3}\% \times \frac{90}{100}x \)
\( = \frac{100}{3 \times 100} \times \frac{90}{100}x \)
\( = \frac{1}{3} \times \frac{90}{100}x \)
\( = \frac{30}{100}x \) (2)
The number of mangoes remaining is the number of good mangoes minus the number of mangoes sold.
From (1) and (2):
\( \frac{90}{100}x - \frac{30}{100}x = 240 \)
\( \implies \frac{90x - 30x}{100} = 240 \)
\( \implies \frac{60x}{100} = 240 \)
\( \implies x = \frac{240 \times 100}{60} \)
\( \implies x = 4 \times 100 \)
\( \implies x = 400 \)
So, initially, he had 400 mangoes. This method helps us track the mangoes at each stage of the sale.
In simple words: First, we find how many mangoes were good. Then, we find how many were sold from the good ones. We subtract the sold mangoes from the good ones, and this number is 240. We use these steps to find the total mangoes bought at the start.

🎯 Exam Tip: Always break down percentage problems into clear steps. Calculate percentages of the original amount first, then the remaining amount, to avoid mistakes.

 

Question 2. A student gets 31 % marks in an examination but fails by 12 marks. If the pass percentage is 35%, find the maximum marks of the examination.
Answer: Let 'x' be the maximum marks possible in the exam. This is the total score one can get.
The pass percentage is given as 35%.
Pass mark \( = \frac{35}{100} \times x = \frac{35x}{100} \)
The student got 31% marks \( = \frac{31}{100} \times x = \frac{31x}{100} \)
The student failed by 12 marks. This means their score was 12 marks less than the pass mark.
So, the student's mark \( = \) Pass mark \( - 12 \)
\( \frac{31x}{100} = \frac{35x}{100} - 12 \)
To solve for x, we rearrange the equation:
\( 12 = \frac{35x}{100} - \frac{31x}{100} \)
\( \implies 12 = \frac{35x - 31x}{100} \)
\( \implies 12 = \frac{4x}{100} \)
\( \implies 4x = 12 \times 100 \)
\( \implies 4x = 1200 \)
\( \implies x = \frac{1200}{4} \)
\( \implies x = 300 \)
The maximum marks for the exam are 300. This calculation helps determine the full value of the exam.
In simple words: We know the pass mark percentage and the student's percentage, and how many marks they failed by. We set up an equation where the pass mark minus 12 is the student's mark, then solve to find the total maximum marks.

🎯 Exam Tip: When dealing with percentages and marks, define 'x' clearly (e.g., maximum marks) and express all marks (pass, student's) in terms of 'x' before setting up the equation.

 

Question 3. Sultana bought the following things from a general store. Calculate the total bill amount paid by her.
(i) Medicines costing Rs 800 with GST at 5%
(ii) Cosmetics costing Rs 650 with GST at 12%
(iii) Cereals costing Rs 900 with GST at 0%
(iv) Sunglass costing Rs 1750 with GST at 18%
(v) Air Conditioner costing Rs 28500 with GST at 28 %
Answer: The formula to calculate the bill amount when GST is applied is: Bill Amount \( = \) Cost \( \times \left(1 + \frac{\text{GST \%}}{100}\right) \)
(i) For Medicines:
Cost \( = \) Rs 800, GST \( = \) 5%
Bill amount \( = 800 \left(1 + \frac{5}{100}\right) = 800 \times \frac{105}{100} = 8 \times 105 = 840 \)
(ii) For Cosmetics:
Cost \( = \) Rs 650, GST \( = \) 12%
Bill amount \( = 650 \left(1 + \frac{12}{100}\right) = 650 \times \frac{112}{100} = 6.5 \times 112 = 728 \)
(iii) For Cereals:
Cost \( = \) Rs 900, GST \( = \) 0%
Bill amount \( = 900 \left(1 + \frac{0}{100}\right) = 900 \times 1 = 900 \)
(iv) For Sunglass:
Cost \( = \) Rs 1750, GST \( = \) 18%
Bill amount \( = 1750 \left(1 + \frac{18}{100}\right) = 1750 \times \frac{118}{100} = 17.5 \times 118 = 2065 \)
(v) For Air Conditioner:
Cost \( = \) Rs 28500, GST \( = \) 28%
Bill amount \( = 28500 \left(1 + \frac{28}{100}\right) = 28500 \times \frac{128}{100} = 285 \times 128 = 36480 \)
Now, we find the total bill amount by adding up all the individual bill amounts.
Total Bill amount \( = 840 + 728 + 900 + 2065 + 36480 \)
\( = 41013 \)
The total bill amount paid by Sultana is Rs 41,013. Calculating each item separately helps manage complex billing.
In simple words: For each item, we add the GST percentage to 100, divide by 100, and multiply by the item's cost to get its final price. Then we add up all the final prices to find the total bill.

🎯 Exam Tip: Always convert percentages to decimals or fractions correctly before calculations. Remember that 0% GST means no extra charge, and the price remains the same.

 

Question 4. If P's income is 25% more than that of Q. By what percentage is Q's income less than P's?
Answer: Let's say Q's income is Rs 100. This is a common starting point for percentage problems.
P's income is 25% more than Q's income.
P's income \( = 100 + \frac{25}{100} \times 100 = 100 + 25 = 125 \)
So, Q's income (100) is 25 less than P's income (125).
To find the percentage by which Q's income is less than P's, we use P's income as the base for comparison.
Percentage less \( = \frac{\text{P's income} - \text{Q's income}}{\text{P's income}} \times 100 \)
\( = \frac{125 - 100}{125} \times 100 \)
\( = \frac{25}{125} \times 100 \)
\( = \frac{1}{5} \times 100 \)
\( = 20\% \)
Therefore, Q's income is 20% less than P's income. It's important to choose the correct base for comparison when calculating percentage differences.
In simple words: We imagine Q earns Rs 100, so P earns Rs 125. Then we calculate how much less Q earns compared to P, but as a percentage of P's earnings.

🎯 Exam Tip: Be careful to use the correct base (the "of what" amount) when calculating percentage increases or decreases. If comparing 'A' to 'B', the base should be 'B'.

 

Question 5. Vaidegi sold two sarees for Rs 2200 each. On one she gains 10% and on the other she loses 12%. Find her total gain or loss percentage in the sale of the sarees.
Answer: For each saree, the selling price (SP) is Rs 2200. We need to find the cost price (CP) for each saree first.
The formula for selling price with gain/loss is: SP \( = \text{CP} \left(1 \pm \frac{\text{Gain/Loss \%}}{100}\right) \)
**Saree 1:**
Selling Price (SP) \( = \) Rs 2200
Gain \( = \) 10%
\( 2200 = \text{CP}_1 \left(1 + \frac{10}{100}\right) \)
\( 2200 = \text{CP}_1 \left(\frac{100+10}{100}\right) \)
\( 2200 = \text{CP}_1 \left(\frac{110}{100}\right) \)
\( \implies \text{CP}_1 = 2200 \times \frac{100}{110} \)
\( \implies \text{CP}_1 = 20 \times 100 \)
\( \implies \text{CP}_1 = 2000 \)
The cost price of Saree 1 is Rs 2000.
**Saree 2:**
Selling Price (SP) \( = \) Rs 2200
Loss \( = \) 12%
\( 2200 = \text{CP}_2 \left(1 - \frac{12}{100}\right) \)
\( 2200 = \text{CP}_2 \left(\frac{100-12}{100}\right) \)
\( 2200 = \text{CP}_2 \left(\frac{88}{100}\right) \)
\( \implies \text{CP}_2 = 2200 \times \frac{100}{88} \)
\( \implies \text{CP}_2 = 25 \times 100 \)
\( \implies \text{CP}_2 = 2500 \)
The cost price of Saree 2 is Rs 2500.

Now, we calculate the total cost price and total selling price.
Total Cost Price \( = \text{CP}_1 + \text{CP}_2 = 2000 + 2500 = 4500 \)
Total Selling Price \( = 2 \times 2200 = 4400 \)
Since the Total Selling Price (Rs 4400) is less than the Total Cost Price (Rs 4500), there is a loss.
Total Loss \( = \) Total Cost Price \( - \) Total Selling Price \( = 4500 - 4400 = 100 \)
Loss Percentage \( = \frac{\text{Loss}}{\text{Total Cost Price}} \times 100 \)
\( = \frac{100}{4500} \times 100 \)
\( = \frac{100}{45} \)
\( = \frac{20}{9} \)
\( = 2\frac{2}{9}\% \)
Vaidegi's total loss percentage in the sale of the sarees is \( 2\frac{2}{9}\% \). Understanding individual profit and loss helps calculate the overall financial outcome.
In simple words: We find the original buying price for each saree. Then we add up the buying prices and the selling prices to see if she made a total profit or loss. Finally, we convert this total loss into a percentage.

🎯 Exam Tip: When calculating total gain or loss percentage for multiple items, always find the *total* cost price and *total* selling price first, then calculate the overall gain/loss percentage based on these totals.

 

Question 6. If 32 men working 12 hours a day can do a work in 15 days, then how many men working 10 hours a day can do double that work in 24 days?
Answer: We can solve this using the formula relating men, days, hours, and work: \( \frac{P_1 \times D_1 \times H_1}{W_1} = \frac{P_2 \times D_2 \times H_2}{W_2} \)
Where P = Men, D = Days, H = Hours, W = Work.
Let's list the given values:
**Initial Situation (1):**
\( P_1 = 32 \) men
\( D_1 = 15 \) days
\( H_1 = 12 \) hours/day
\( W_1 = 1 \) unit of work (we assume one unit of work)

**New Situation (2):**
\( P_2 = x \) men (what we need to find)
\( D_2 = 24 \) days
\( H_2 = 10 \) hours/day
\( W_2 = 2 \) units of work (double the original work)

Now, we put these values into the formula:

\( \frac{32 \times 15 \times 12}{1} = \frac{x \times 24 \times 10}{2} \)
\( \implies 32 \times 15 \times 12 = x \times 12 \times 10 \)
\( \implies x = \frac{32 \times 15 \times 12}{12 \times 10} \)
\( \implies x = \frac{32 \times 15}{10} \)
\( \implies x = 32 \times \frac{3}{2} \)
\( \implies x = 16 \times 3 \)
\( \implies x = 48 \)
So, 48 men working 10 hours a day can do double the work in 24 days. This type of problem highlights how different factors like manpower and work duration are interconnected.
In simple words: We use a formula that connects how many men, days, and hours are needed for a certain amount of work. We plug in what we know and what we want to find (double the work) to figure out how many men are needed.

🎯 Exam Tip: Remember to use the formula \( \frac{P_1 D_1 H_1}{W_1} = \frac{P_2 D_2 H_2}{W_2} \) for "work and time" problems. Make sure to correctly account for the change in the amount of work if specified (e.g., "double the work").

 

Question 7. Amutha can weave a saree in 18 days. Anjali is twice as good a weaver as Amutha. If both of them weave together, then in how many days can they complete weaving the saree?
Answer: First, let's find the individual work rates. Anjali is twice as efficient as Amutha, meaning she works faster.
Amutha takes 18 days to weave a saree.
Since Anjali is twice as good, she will take half the time Amutha takes.
Time taken by Anjali \( = \frac{18}{2} = 9 \) days.
Now, we need to find the time taken when they work together. We can use the formula for two people working together: Time \( = \frac{\text{Time A} \times \text{Time B}}{\text{Time A} + \text{Time B}} \)
Let Amutha's time be 'a' and Anjali's time be 'b'.
Time taken by them together \( = \frac{a \times b}{a + b} \)
\( = \frac{18 \times 9}{18 + 9} \)
\( = \frac{162}{27} \)
\( = 6 \)
So, they can complete weaving the saree in 6 days if they work together. Working as a team significantly reduces the total time needed for a task.
In simple words: We know how long Amutha takes. Anjali is twice as fast, so she takes half the time. We use a special formula for two people working together to find how many days they need if they both weave at the same time.

🎯 Exam Tip: For "work and time" problems involving two individuals, remember the combined time formula \( \frac{T_1 \times T_2}{T_1 + T_2} \). Also, clearly understand what "twice as good" means in terms of time taken.

 

Question 8. P and Q can do a piece of work in 12 days and 15 days respectively. P started the work alone and then after 3 days, Q joined him till the work was completed. How long did the work last?
Answer: We need to calculate how much work P does alone, then the remaining work, and finally how long P and Q take to finish it together.
P can do a piece of work in 12 days.
P's 1 day work \( = \frac{1}{12} \)
P worked alone for 3 days.
P's 3 days work \( = 3 \times \frac{1}{12} = \frac{3}{12} \)
Remaining work after 3 days \( = 1 - \frac{3}{12} = \frac{12 - 3}{12} = \frac{9}{12} \)
Q can do a piece of work in 15 days.
Q's 1 day work \( = \frac{1}{15} \)
Now, P and Q work together to finish the remaining work.
(P + Q)'s 1 day work \( = \frac{1}{12} + \frac{1}{15} \)
To add these fractions, we find a common denominator, which is 60.
\( = \frac{5}{60} + \frac{4}{60} = \frac{5+4}{60} = \frac{9}{60} \)
Number of days required to finish the remaining work \( = \frac{\text{Remaining work}}{\text{(P+Q)'s 1 day work}} \)
\( = \frac{\frac{9}{12}}{\frac{9}{60}} \)
\( = \frac{9}{12} \times \frac{60}{9} \)
\( = \frac{60}{12} \)
\( = 5 \) days.
So, P and Q together finished the remaining work in 5 days.
The total work lasted for the 3 days P worked alone plus the 5 days they worked together.
Total work lasted \( = 3 + 5 = 8 \) days. Breaking down the work into phases simplifies tracking progress.
In simple words: First, we find out how much work P did by himself. Then, we see how much work is left. Next, we figure out how much work P and Q can do together in one day. Finally, we divide the remaining work by their combined daily work to find how many more days it took, and add that to the initial days.

🎯 Exam Tip: In work-and-time problems where people join midway, calculate the work done by the first person(s) and subtract it from the total work (usually 1). Then, calculate the combined work rate for the remaining part.

 

Question 9. If the numerator of a fraction is increased by 50% and the denominator is decreased by 20%, then it becomes \( \frac{3}{5} \). Find the original fraction.
Answer: Let the original fraction be \( \frac{x}{y} \).
The numerator is increased by 50%.
New Numerator \( = x + \frac{50}{100}x = x + 0.5x = 1.5x = \frac{150}{100}x \)
The denominator is decreased by 20%.
New Denominator \( = y - \frac{20}{100}y = y - 0.2y = 0.8y = \frac{80}{100}y \)
The new fraction formed is \( \frac{3}{5} \).
So, we can write the equation:
\( \frac{\frac{150}{100}x}{\frac{80}{100}y} = \frac{3}{5} \)
\( \implies \frac{150x}{80y} = \frac{3}{5} \)
\( \implies \frac{15x}{8y} = \frac{3}{5} \)
Now, we solve for \( \frac{x}{y} \):
\( \frac{x}{y} = \frac{3}{5} \times \frac{8}{15} \)
\( \implies \frac{x}{y} = \frac{3 \times 8}{5 \times 15} \)
\( \implies \frac{x}{y} = \frac{1 \times 8}{5 \times 5} \)
\( \implies \frac{x}{y} = \frac{8}{25} \)
The original fraction is \( \frac{8}{25} \). This process of setting up equations for changes in numerator and denominator is key to finding the original fraction.
In simple words: We set up the original fraction as x over y. We calculate the new numerator by adding 50% to x, and the new denominator by taking away 20% from y. We put these new values into a fraction and set it equal to 3/5, then solve to find the original x over y.

🎯 Exam Tip: When dealing with percentage increases or decreases, remember to multiply the original value by \( (1 + \text{percentage}/100) \) for increase and \( (1 - \text{percentage}/100) \) for decrease.

 

Question 10. Gopi sold a laptop at 12% gain. If it had been sold for Rs 1200 more, the gain would have been 20%. Find the cost price of the laptop.
Answer: Let 'x' be the cost price (CP) of the laptop. This is the base value we need to find.
First Selling Price (SP1) with 12% gain:
\( \text{SP}_1 = \text{CP} \left(1 + \frac{\text{Gain \%}}{100}\right) \)
\( \text{SP}_1 = x \left(1 + \frac{12}{100}\right) \)
\( \text{SP}_1 = x \left(\frac{100+12}{100}\right) = \frac{112}{100}x \)
If it had been sold for Rs 1200 more, the gain would have been 20%.
New Selling Price (SP2) \( = \text{SP}_1 + 1200 \)
Also, with 20% gain, the new selling price would be:
\( \text{SP}_2 = x \left(1 + \frac{20}{100}\right) \)
\( \text{SP}_2 = x \left(\frac{100+20}{100}\right) = \frac{120}{100}x \)
Now, we can set up the equation:
\( \frac{112}{100}x + 1200 = \frac{120}{100}x \)
To solve for x, we rearrange the terms:
\( 1200 = \frac{120}{100}x - \frac{112}{100}x \)
\( \implies 1200 = \frac{120x - 112x}{100} \)
\( \implies 1200 = \frac{8x}{100} \)
\( \implies 8x = 1200 \times 100 \)
\( \implies 8x = 120000 \)
\( \implies x = \frac{120000}{8} \)
\( \implies x = 15000 \)
The cost price of the laptop is Rs 15,000. Comparing different selling scenarios helps determine the original cost.
In simple words: We write down two different selling prices based on two different profit percentages. One selling price is the first one, plus Rs 1200. We set these two expressions for the second selling price equal to each other and solve for the laptop's original cost.

🎯 Exam Tip: Always represent the cost price as 'x' and form two separate equations for the selling price based on the given gain percentages and additional amount. Then equate or subtract them to find 'x'.

 

Question 11. A shopkeeper gives two successive discounts on an article whose marked price is Rs 180 and selling price is Rs 108. Find the first discount percentage if the second discount is 25%.
Answer: Let \( d_1 \) be the first discount percentage and \( d_2 \) be the second discount percentage.
Marked Price (MP) \( = \) Rs 180
Selling Price (SP) after two discounts \( = \) Rs 108
Second discount \( d_2 = 25\% \)
The formula for successive discounts is: SP \( = \text{MP} \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \)
Substitute the known values:
\( 108 = 180 \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{25}{100}\right) \)
\( \implies 108 = 180 \times \left(1 - \frac{d_1}{100}\right) \times \left(\frac{100-25}{100}\right) \)
\( \implies 108 = 180 \times \left(1 - \frac{d_1}{100}\right) \times \left(\frac{75}{100}\right) \)
\( \implies 108 = 180 \times \left(1 - \frac{d_1}{100}\right) \times \frac{3}{4} \)
Now, we isolate the term with \( d_1 \):
\( 1 - \frac{d_1}{100} = \frac{108 \times 4}{180 \times 3} \)
\( \implies 1 - \frac{d_1}{100} = \frac{432}{540} \)
\( \implies 1 - \frac{d_1}{100} = \frac{4}{5} \)
Now, solve for \( d_1 \):
\( \frac{d_1}{100} = 1 - \frac{4}{5} \)
\( \implies \frac{d_1}{100} = \frac{5 - 4}{5} \)
\( \implies \frac{d_1}{100} = \frac{1}{5} \)
\( \implies d_1 = \frac{1}{5} \times 100 \)
\( \implies d_1 = 20 \)
The first discount percentage is 20%. Calculating discounts in reverse helps find missing information about pricing strategies.
In simple words: We use a formula that calculates the final price after two discounts. We know the original price, the final price, and the second discount. We plug these numbers into the formula and solve to find the first discount percentage.

🎯 Exam Tip: When dealing with successive discounts, apply them one after another to the remaining value, not the original marked price. The formula \( \text{SP} = \text{MP} \times (1 - d_1/100) \times (1 - d_2/100) \) is a shortcut for this.

 

Question 12. Find the rate of compound interest at which a principal becomes 1.69 times itself in 2 years.
Answer: Let the principal amount be 'P'.
The amount (A) after compound interest is given as 1.69 times the principal.
So, \( A = 1.69 P \).
The time period (n) is 2 years.
We need to find the rate of interest (r).
The formula for compound interest is: \( A = P \left(1 + \frac{r}{100}\right)^n \)
Substitute the known values into the formula:
\( 1.69 P = P \left(1 + \frac{r}{100}\right)^2 \)
Divide both sides by P:
\( 1.69 = \left(1 + \frac{r}{100}\right)^2 \)
To remove the square, take the square root on both sides:
\( \sqrt{1.69} = 1 + \frac{r}{100} \)
\( \implies 1.3 = 1 + \frac{r}{100} \)
Now, solve for r:
\( \frac{r}{100} = 1.3 - 1 \)
\( \implies \frac{r}{100} = 0.3 \)
\( \implies r = 0.3 \times 100 \)
\( \implies r = 30 \)
The rate of compound interest is 30%. This calculation shows how quickly an investment grows over time with compound interest.
In simple words: We know the final amount is 1.69 times the start amount after 2 years. Using the compound interest formula, we plug in these values and solve for the interest rate by taking the square root.

🎯 Exam Tip: Remember that when the principal becomes 'x' times itself, you can directly set \( \left(1 + \frac{r}{100}\right)^n = x \). Taking square roots or cube roots is a common step to simplify the equation.

 

Question 13. A small - scale company undertakes an agreement to make 540 motor pumps in 150 days and employs 40 men for the work. After 75 days, the company could make only 180 motor pumps. How many more men should the company employ so that the work is completed on time as per the agreement?
Answer: We will use the concept of men-days-work relationship. The total work needs to be completed in 150 days.
**Planned Work:**
Total Motor pumps \( = 540 \)
Total Days \( = 150 \)
Men employed \( = 40 \)

**Work Done After 75 Days:**
Days passed \( = 75 \)
Motor pumps made \( = 180 \)
Remaining pumps to be made \( = 540 - 180 = 360 \)
Remaining days \( = 150 - 75 = 75 \)
Let 'x' be the additional men needed.
Total men required for remaining work \( = 40 + x \)

We can use the formula: \( \frac{P_1 D_1}{W_1} = \frac{P_2 D_2}{W_2} \)
Using the initial 75 days for 180 pumps and the next 75 days for 360 pumps:
\( P_1 = 40 \), \( D_1 = 75 \), \( W_1 = 180 \)
\( P_2 = 40 + x \), \( D_2 = 75 \), \( W_2 = 360 \)
\( \frac{40 \times 75}{180} = \frac{(40 + x) \times 75}{360} \)
We can cancel 75 from both sides:
\( \frac{40}{180} = \frac{40 + x}{360} \)
\( \implies \frac{40}{180} = \frac{40 + x}{2 \times 180} \)
\( \implies 40 = \frac{40 + x}{2} \)
\( \implies 40 \times 2 = 40 + x \)
\( \implies 80 = 40 + x \)
\( \implies x = 80 - 40 \)
\( \implies x = 40 \)
The company should employ 40 more men to complete the work on time. This problem illustrates how adjustments in manpower can ensure project deadlines are met.
In simple words: We calculate how many pumps were made in the first half of the time. Then we see how many pumps are left and how many days are left. Using the relationship between men, days, and work, we figure out how many total men are needed for the rest of the work and subtract the existing men to find the extra men needed.

🎯 Exam Tip: For problems involving changes in men, days, or work, it's best to use the formula \( \frac{\text{Men}_1 \times \text{Days}_1}{\text{Work}_1} = \frac{\text{Men}_2 \times \text{Days}_2}{\text{Work}_2} \). Be careful to correctly identify the 'remaining work' and 'remaining days'.

 

Question 14. P alone can do \( \frac{1}{2} \) of a work in 6 days and Q alone can do \( \frac{2}{3} \) of the same work in 4 days. In how many days will they finish \( \frac{3}{4} \) of the work, working together?
Answer: First, let's find the time each person takes to complete the full work.
P can do \( \frac{1}{2} \) of the work in 6 days.
So, P will do the full work (1 unit) in \( 6 \times 2 = 12 \) days.
Q can do \( \frac{2}{3} \) of the work in 4 days.
So, Q will do the full work (1 unit) in \( 4 \times \frac{3}{2} = 2 \times 3 = 6 \) days.

Now, let's find their combined work rate. If P takes 'a' days and Q takes 'b' days to complete the full work, working together they take \( \frac{a \times b}{a + b} \) days.
Combined time to complete full work \( = \frac{12 \times 6}{12 + 6} \)
\( = \frac{72}{18} \)
\( = 4 \) days.
So, P and Q together can finish the whole work in 4 days. We need to find how many days they will take to finish \( \frac{3}{4} \) of the work.
Time to finish \( \frac{3}{4} \) of the work \( = \) (Combined time for full work) \( \times \frac{3}{4} \)
\( = 4 \times \frac{3}{4} \)
\( = 3 \) days.
They will finish \( \frac{3}{4} \) of the work in 3 days. Calculating individual efficiency first helps determine collective progress.
In simple words: First, we figure out how many days P would take to do all the work and how many days Q would take to do all the work. Then, we find out how many days they would take if they worked together on the whole job. Finally, we take three-fourths of that time to find how long it takes for three-fourths of the work.

🎯 Exam Tip: Always convert fractional work rates to "time taken for full work" for each individual first. Then, use combined work formulas and finally adjust for the required fraction of work.

 

Question 15. X alone can do a piece of work in 6 days and Y alone in 8 days. X and Y undertook the work for Rs 48000. With the help of Z, they completed the work in 3 days. Find Z's share?
Answer: The share of each person is proportional to the amount of work they do.
X can do the work in 6 days.
X's 1 day work \( = \frac{1}{6} \)
Y can do the work in 8 days.
Y's 1 day work \( = \frac{1}{8} \)
With the help of Z, they completed the work in 3 days.
So, in 3 days, the work done by X \( = 3 \times \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \) of the work.
In 3 days, the work done by Y \( = 3 \times \frac{1}{8} = \frac{3}{8} \) of the work.
Total work done by X and Y together in 3 days \( = \frac{1}{2} + \frac{3}{8} \)
To add these, find a common denominator, which is 8.
\( = \frac{4}{8} + \frac{3}{8} = \frac{7}{8} \) of the work.
The remaining work was done by Z in 3 days.
Work done by Z \( = 1 - \frac{7}{8} = \frac{1}{8} \) of the work.

The total amount for the work is Rs 48,000.
X's share \( = \frac{1}{2} \times 48000 = 24000 \)
Y's share \( = \frac{3}{8} \times 48000 = 3 \times 6000 = 18000 \)
Z's share \( = \frac{1}{8} \times 48000 = 6000 \)
Alternatively, Z's share can be calculated as total amount minus X's and Y's shares.
Z's share \( = 48000 - (24000 + 18000) = 48000 - 42000 = 6000 \)
Z's share is Rs 6,000. Dividing the payment based on the work done by each individual ensures fairness.
In simple words: First, we find how much work X and Y each do in the 3 days. Then we see how much work is left, which Z must have done. Each person's share of the Rs 48,000 is then calculated based on the fraction of the total work they completed.

🎯 Exam Tip: In work-and-wages problems, the payment received by each person is directly proportional to the amount of work they complete. Always calculate the fraction of work done by each individual first.

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