Explanation for the Arithmetic Quiz 07

This is the detailed explanation for the Arithmetic Quiz 07. Each question’s solution is broken down step-by-step to help you understand the core concepts and formulas.

---

1. A person sold a horse at a gain of 15%. Had he bought it for 25% less and sold it for Rs. 600 less, he would have made a profit of 32%. The cost price of the horse was:

Answer: Rs. 3,750

• Step 1: Define the initial variables.
  • Let the original cost price (CP) be $x$.
  • Original selling price (SP) = CP + 15% of CP = $x+0.15x=1.15x$.
• Step 2: Define the new scenario.
  • New CP = Original CP – 25% of Original CP = $x-0.25x=0.75x$.
  • New SP = Original SP – 600 = $1.15x-600$.
  • New profit = 32% of New CP = $0.32\times 0.75x=0.24x$.
• Step 3: Set up the equation and solve for x.
  • New SP = New CP + New Profit
  • $1.15x-600=0.75x+0.24x$
  • $1.15x-600=0.99x$
  • $1.15x-0.99x=600$
  • $0.16x=600$
  • $x=600/0.16=60000/16=\text{Rs. }3750$.
  • The cost price of the horse was Rs. 3,750.

---

2. By selling a bicycle for Rs. 2,850, a shopkeeper gains 14%. If the profit is reduced to 8%, then the selling price will be:

Answer: Rs. 2,700

• Step 1: Find the original cost price (CP).
  • SP = $2850$, Profit % = 14%.
  • $CP = (SP \times 100) / (100 + \text{Profit %})$
  • $CP=(2850\times 100)/(100+14)=285000/114=\text{Rs. }2500$.
• Step 2: Calculate the new selling price (SP) for an 8% profit.
  • New SP = CP + 8% of CP
  • New SP = $2500+(0.08\times 2500)=2500+200=\text{Rs. }2700$.
  • The new selling price will be Rs. 2,700.

---

3. A shopkeeper earns a profit of 12% on selling a book at 10% discount on the printed price. The ratio of the cost price and the printed price of the book is:

Answer: 45:56

• Step 1: Define the variables.
  • Let the printed price (or marked price, MP) be $x$.
• Step 2: Express the selling price (SP) in terms of the printed price.
  • Discount = 10% of MP = $0.10x$.
  • SP = MP – Discount = $x-0.10x=0.90x$.
• Step 3: Express the selling price in terms of the cost price (CP) and profit.
  • SP = CP + 12% of CP = $1.12\times CP$.
• Step 4: Set the two expressions for SP equal to each other and find the ratio of CP to MP.
  • $1.12\times CP=0.90\times MP$
  • $CP/MP=0.90/1.12=90/112$.
  • Simplify the ratio by dividing by 2: $45/56$.
  • The ratio of cost price to printed price is 45:56.

---

4. If a man were to sell his chair for Rs. 720, he would lose 25%. To gain 25% he should sell it for:

Answer: Rs. 1,200

• Step 1: Find the cost price (CP) of the chair.
  • SP = $720$, Loss % = 25%.
  • CP = $(SP \times 100) / (100 – \text{Loss %})$
  • CP = $(720\times 100)/(100-25)=72000/75=\text{Rs. }960$.
• Step 2: Calculate the new selling price (SP) for a 25% gain.
  • New SP = CP + 25% of CP
  • New SP = $960+(0.25\times 960)=960+240=\text{Rs. }1200$.
  • He should sell it for Rs. 1,200.

---

5. If the profit per cent got on selling an article is numerically equal to its cost price in rupees and the selling price is Rs. 39, then cost price (in Rs.) will be:

Answer: 30

• Step 1: Set up the equation.
  • Let the cost price (CP) be $x$.
  • The profit % is also $x$.
  • SP = CP + Profit = x+(x/100)×x=x+x2/100.
• Step 2: Plug in the selling price and solve the quadratic equation.
  • 39=x+x2/100
  • 3900=100x+x2
  • x2+100x−3900=0.
  • We can factor this as $(x+130)(x-30)=0$.
  • The cost price cannot be negative, so $x=30$.
• Step 3: Verify the answer.
  • If CP = Rs. 30, the profit % is 30%.
  • Profit amount = 30% of $30=0.30\times 30=9$.
  • SP = CP + Profit = $30+9=39$. This matches the given information. The cost price is Rs. 30.

---

6. A man sold two chairs at Rs. 1,200 each. On one he gained 20% and on the other he lost 20%. His gain or loss in the whole transaction is:

Answer: 4% loss

• Shortcut: When two items are sold at the same price, and one is sold at a gain of $x\%$ and the other at a loss of $x\%$, there will always be a net loss.
  • The loss percentage is given by the formula: (x/10)2%.
  • Here, $x=20$.
  • Loss % = (20/10)2=22=4%.
  • The loss is 4%.
• Detailed calculation:
  • Chair 1 (20% gain): SP = $1200$. CP1 = $(1200\times 100)/(100+20)=120000/120=\text{Rs. }1000$.
  • Chair 2 (20% loss): SP = $1200$. CP2 = $(1200\times 100)/(100-20)=120000/80=\text{Rs. }1500$.
  • Total CP: $1000+1500=\text{Rs. }2500$.
  • Total SP: $1200+1200=\text{Rs. }2400$.
  • Total Loss: $2500-2400=\text{Rs. }100$.
  • Loss %: $($Loss / Total CP) $\times 100\%=(100/2500)\times 100\%=(1/25)\times 100\%=4\%$.

---

7. An article is listed at Rs. 920. A customer pays Rs. 742.90 for it after getting two successive discounts. If the rate of first discount is 15%, the rate of 2nd discount is:

Answer: 5%

• Step 1: Calculate the price after the first discount.
  • Listed Price = Rs. 920. First Discount = 15%.
  • Price after 1st discount = $920-(0.15\times 920)=920-138=\text{Rs. }782$.
• Step 2: Calculate the amount of the second discount.
  • Price paid = Rs. 742.90.
  • 2nd Discount amount = Price after 1st discount – Price paid = $782-742.90=\text{Rs. }39.10$.
• Step 3: Calculate the rate of the second discount. The discount is applied to the price after the first discount.
  • 2nd Discount % = (2nd Discount amount / Price after 1st discount) $\times 100\%$
  • 2nd Discount % = $(39.10/782)\times 100\%=0.05\times 100\%=5\%$.
  • The rate of the 2nd discount is 5%.

---

8. A man buys a field of agricultural land for Rs. 3,60,000. He sells one-third at a loss of 20% and two-fifths at a gain of 25%. At what price must he sell the remaining field so as to make an overall profit of 10%?

Answer: Rs. 1,20,000

• Step 1: Calculate the overall target selling price.
  • Total CP = Rs. 3,60,000. Overall Profit % = 10%.
  • Overall SP = Total CP + 10% of Total CP = $360000+(0.10\times 360000)=360000+36000=\text{Rs. }3,96,000$.
• Step 2: Calculate the CP and SP of the first two parts.
  • Part 1 (one-third):
    • CP = $(1/3)\times 360000=\text{Rs. }1,20,000$.
    • Loss = 20% of $120000=0.20\times 120000=\text{Rs. }24,000$.
    • SP = $120000-24000=\text{Rs. }96,000$.
  • Part 2 (two-fifths):
    • CP = $(2/5)\times 360000=\text{Rs. }1,44,000$.
    • Gain = 25% of $144000=0.25\times 144000=\text{Rs. }36,000$.
    • SP = $144000+36000=\text{Rs. }1,80,000$.
• Step 3: Calculate the CP of the remaining field.
  • Remaining fraction = $1-1/3-2/5=(15-5-6)/15=4/15$.
  • CP of remaining field = $(4/15)\times 360000=4\times 24000=\text{Rs. }96,000$.
• Step 4: Calculate the required SP for the remaining field.
  • Required SP = Overall SP – (SP of Part 1 + SP of Part 2)
  • Required SP = $396000-(96000+180000)=396000-276000=\text{Rs. }1,20,000$.
  • He must sell the remaining field for Rs. 1,20,000.

---

9. A shopkeeper marks his goods 30% above his cost price but allows a discount of 10% at the time of sale. His gain is:

Answer: 17%

• Step 1: Assume a cost price. Let CP = Rs. 100.
• Step 2: Calculate the marked price (MP).
  • MP = CP + 30% of CP = $100+30=\text{Rs. }130$.
• Step 3: Calculate the selling price (SP) after the discount.
  • Discount = 10% of MP = $0.10\times 130=\text{Rs. }13$.
  • SP = MP – Discount = $130-13=\text{Rs. }117$.
• Step 4: Calculate the gain percentage.
  • Gain = SP – CP = $117-100=\text{Rs. }17$.
  • Gain % = (Gain / CP) $\times 100\%=(17/100)\times 100\%=17\%$.
  • His gain is 17%.

---

10. A sells an article to B at a profit of 10% B sells the article back to A at a loss of 10%. In this transaction:

Answer: A makes a profit of 11%

• Step 1: Assume a cost price for A.
  • Let the original CP for A be Rs. 100.
• Step 2: Calculate the price B paid for the article.
  • A sells to B at a 10% profit.
  • CP for B = A’s CP + 10% of A’s CP = $100+10=\text{Rs. }110$.
• Step 3: Calculate the price A paid to buy the article back from B.
  • B sells back to A at a 10% loss. The loss is on B’s cost price.
  • Loss for B = 10% of B’s CP = $0.10\times 110=\text{Rs. }11$.
  • A’s new CP (B’s SP) = B’s CP – B’s Loss = $110-11=\text{Rs. }99$.
• Step 4: Calculate A’s total profit or loss.
  • A’s initial expenditure = Rs. 100.
  • A’s final income from the transaction = Rs. 110.
  • Wait, the question asks about the whole transaction. A buys the article for 100, sells it for 110. A then buys it back for 99.
  • Total money A spent = Rs. 100 (initially) + Rs. 99 (to buy it back) = Rs. 199.
  • Total money A received = Rs. 110 (from selling to B).
  • Wait, this interpretation is wrong. The transaction is a single chain. A sells to B, then B sells back to A.
  • A’s gain is the difference between what he sold it for (to B) and what he bought it back for (from B).
  • A’s SP to B = $100\times 1.10=110$.
  • A’s CP from B = $110\times 0.90=99$.
  • A’s total profit = Price A sold for – Price A bought back for = $110-99=\text{Rs. }11$.
  • A’s profit percentage = (Profit / A’s initial CP) $\times 100\%=(11/100)\times 100\%=11\%$.
  • A makes a profit of 11%.