Question

To construct a wall, sand and cement are mixed in the ratio of 3:1. The cost of sand and that of cement are in the ratio of 1:2.
If the total cost of sand and cement to construct the wall is 1000 rupees, then what is the cost (in rupees) of cement used?

• A. 400
• B. 600
• C. 800
• D. 200

Hint:

When a problem gives a ratio of quantities (A:B) and a ratio of unit prices (C:D), the ratio of total costs will be (A\(\times\)C) : (B\(\times\)D). In this case, (3\(\times\)1) : (1\(\times\)2) = 3:2.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves combining two different ratios: a ratio of quantities and a ratio of costs per unit. The goal is to find the ratio of the total costs of the components and then use it to find the actual cost of one component.
Step 2: Key Formula or Approach:
Total Cost of a component = (Quantity of the component) \(\times\) (Cost per unit of the component).
We need to find the ratio of (Total Cost of Sand) : (Total Cost of Cement).
Step 3: Detailed Calculation:
Let the common multiplier for the quantity ratio be \(x\) and for the cost ratio be \(y\).
Ratio of Quantities:
Sand : Cement = 3 : 1 So, let the quantity of sand used be \(3x\) units and the quantity of cement used be \(1x\) units.
Ratio of Costs per Unit:
Cost of Sand : Cost of Cement = 1 : 2 So, let the cost per unit of sand be \(1y\) rupees and the cost per unit of cement be \(2y\) rupees.
Calculate the Total Cost for each component:
Total Cost of Sand = (Quantity of Sand) \(\times\) (Cost per unit of Sand) = \((3x) \times (1y) = 3xy\)
Total Cost of Cement = (Quantity of Cement) \(\times\) (Cost per unit of Cement) = \((1x) \times (2y) = 2xy\)
Find the Ratio of Total Costs:
Ratio of Total Cost of Sand to Total Cost of Cement = \(3xy : 2xy\). The \(xy\) term cancels out, so the ratio of their total costs is 3 : 2.
Calculate the Actual Cost of Cement:
The total cost of the mixture is 1000 rupees. This amount is divided between sand and cement in the ratio 3:2.
Total parts in the ratio = 3 + 2 = 5 parts.
The value of one part = Total Cost / Total Parts = \(1000 / 5 = 200\) rupees.
Cost of Cement = (Cement's share in the ratio) \(\times\) (Value of one part)
Cost of Cement = \(2 \times 200 = 400\) rupees.
Step 4: Final Answer:
The cost of cement used is 400 rupees.
Step 5: Why This is Correct:
The solution correctly calculates the ratio of the total costs by multiplying the quantity ratio by the unit cost ratio. It then uses this final ratio (3:2) to partition the total given cost of 1000 rupees, accurately finding the cost of the cement. (Cost of Sand would be 3 parts = \(3 \times 200 = 600\), and \(600 + 400 = 1000\)).