4 Marks
36
A student entrepreneur started a company that manufactures sanitizers in two sizes: small and large. The cost of a small bottle of sanitizer is ₹10 and that of a large bottle is ₹15. In June, the company sold 1000 bottles and recorded a total sale of ₹12,750. Seeing the increased demand, the company decided to increase the price of both sanitizer bottles by ₹2 each. In the next month, the company sold 2500 bottles and recorded a total sale of ₹34,250.
(i) Form a linear equation in two variables representing the sale for June. [1 Mark]
(ii) Form a linear equation in two variables representing the sale for July. [1 Mark]
(iii)
(A) How many sanitizer bottles of each type were actually sold in June? [2 Marks]
OR (B) How many sanitizer bottles of each type were sold in July?
OR (B) How many sanitizer bottles of each type were sold in July?
Concept:
1
This is an application of Pairs of Linear Equations in Two Variables. Let the number of small bottles sold be $x$ and large bottles be $y$. We form two types of equations for a given month: an "Items Equation" ($x + y = \text{Total Bottles}$) and a "Value Equation" ($(\text{Price}_s \times x) + (\text{Price}_l \times y) = \text{Total Revenue}$). By solving these simultaneous equations via substitution or elimination, we find exact quantities.
Hint:
1
For (i): June prices are ₹10 and ₹15. Revenue is ₹12750.
2
For (ii): July prices increased by ₹2, making them ₹12 and ₹17. Revenue is ₹34250.
3
For (iii) A: Use June item equation $x + y = 1000$ alongside the equation from part (i). Multiply the item equation by 10 and subtract.
4
For (iii) B: Use July item equation $x + y = 2500$ alongside the equation from part (ii). Multiply the item equation by 12 and subtract.
Mistakes to Avoid:
!
Forgetting the price increase: In part (ii), if you write $10x + 15y = 34250$, you completely missed the "increase the price by ₹2 each" detail. The new prices are 12 and 17.
!
Mixing up the months in part (iii): Ensure you use $x+y=1000$ for June, and $x+y=2500$ for July.
Official CBSE Solution:
Let the number of small bottles be '$x$' and large bottles be '$y$'.
Part (i): June Sales Equation
Prices: Small = ₹10, Large = ₹15. Revenue = ₹12750.
$10x + 15y = 12750$ (or simplified: $2x + 3y = 2550$)
Part (ii): July Sales Equation
Prices increased by ₹2: Small = ₹12, Large = ₹17. Revenue = ₹34250.
$12x + 17y = 34250$
Part (iii) Option A: June Quantities
According to the question, total bottles in June = 1000:
$x + y = 1000$ --- (Eq 1)
$10x + 15y = 12750$ --- (Eq 2)
Multiply (Eq 1) by 10: $10x + 10y = 10000$. Subtract from (Eq 2):
$5y = 2750 \implies y = 550$
Substitute $y$: $x + 550 = 1000 \implies x = 450$
Ans (A): Small = 450 bottles, Large = 550 bottles.
OR Part (iii) Option B: July Quantities
Total bottles in July = 2500:
$x + y = 2500$ --- (Eq 1)
$12x + 17y = 34250$ --- (Eq 2)
Multiply (Eq 1) by 12: $12x + 12y = 30000$. Subtract from (Eq 2):
$5y = 4250 \implies y = 850$
Substitute $y$: $x + 850 = 2500 \implies x = 1650$
Ans (B): Small = 1650 bottles, Large = 850 bottles.
🎯 CBSE Marking Scheme (4 Marks):
• (i) Correct June equation ($10x+15y=12750$): 1 Mark
• (ii) Correct July equation ($12x+17y=34250$): 1 Mark
• (iii) Forming the quantity equation ($x+y=1000$ or $2500$): 1 Mark
• (iii) Solving equations to find correct $x$ and $y$ values: 1 Mark
• (i) Correct June equation ($10x+15y=12750$): 1 Mark
• (ii) Correct July equation ($12x+17y=34250$): 1 Mark
• (iii) Forming the quantity equation ($x+y=1000$ or $2500$): 1 Mark
• (iii) Solving equations to find correct $x$ and $y$ values: 1 Mark
