Question

The cost of the Red, Green and Blue colours pe 0, Rs. 15 and Rs. 18 respectively. Rang Mahal is a renowned building in which these three colours are being used in the ratio of \(3: 2: 4\). The average cost of all the three colours used per kg is 7 (a) 18 (b) 20 (c) \(17.66\) (d) can't be determined

Short answer

(a) 11.00 (b) 11.25 (c) 11.50 (d) can't be determined Answer: (d) can't be determined

Step-by-step solution

Write down the given information

We have: - Cost of Red colour: Rs. 0 - Cost of Green colour: Rs. 15 - Cost of Blue colour: Rs. 18 - Ratio of colours used in Rang Mahal: \(3: 2: 4\)

Calculate the weighted average cost per kg

Using the weighted average formula, which is given by: \(\text{Weighted average} = \frac{\text{Sum of products of value and weight}}{\text{Sum of weights}}\) To find the weighted average, we'll first find the sum of the products of the cost and ratio, and then divide it by the sum of the ratio values: \(\text{Weighted average} = \frac{(0 \times 3) + (15 \times 2) + (18 \times 4)}{3 + 2 + 4}\)

Calculate the sum of the products and the sum of the weights

We have: - Sum of the products: \((0 \times 3) + (15 \times 2) + (18 \times 4) = 0 + 30 + 72 = 102\) - Sum of the weights: \(3 + 2 + 4 = 9\)

Calculate the weighted average cost per kg using the values from Step 3

The weighted average cost per kg is: \(\text{Weighted average} = \frac{102}{9} = 11.333\) Since our choices are in the form of whole numbers or two decimal places, we rounded the result to two decimal places: \(\text{Weighted average cost per kg} = 11.33\) However, none of the given options match our calculated result. So the answer is: (d) can't be determined

Key concepts

Weighted Average Calculation

Weighted average calculation is a method used to find the average in situations where different items contribute unequally to the result. Unlike a simple average, where each item is treated equally, a weighted average takes into account the importance or frequency of each item. In our exercise, we have three colors with different costs and usage ratios.

The formula for weighted average is given as: \[\text{Weighted average} = \frac{\text{Sum of products of value and weight}}{\text{Sum of weights}}\]Here's how you can break down the calculation:

• For each color, multiply its cost by the respective ratio provided. These products represent the 'weight' each color adds to the average.
• Add up all these products to get a total sum.
• Then, calculate the sum of the ratios themselves.
• Finally, divide the total sum of products by the sum of ratios to find the weighted average.

In this problem, the weighted average gives us an adjusted value that reflects the high usage of blue color in the final cost.

Ratio and Proportion

Ratio and proportion are mathematical tools that help in comparing quantities, expressing how one quantity relates to another, or finding out how a set of numbers stand in relation to each other. In our exercise, different colors have been used in a specific ratio: \(3: 2: 4\).

Ratios show the relative size of two or more quantities, whereas proportion is about setting two ratios equal. For instance, saying colors are used in a \(3: 2: 4\) ratio means that for every 3 units of red, 2 units of green and 4 units of blue are used.

• This ratio helps us understand the significance of each color in context to others.
• It also aids in understanding how changes in one quantity will affect others.
• A critical aspect of ratios is that they have no units, making them pure numbers that compare sizes directly.

Understanding ratio and proportion equips students to apply these principles in various real-life scenarios, such as mixing solutions, comparing speeds, or even creating recipes.

Mathematical Problem Solving

Mathematical problem solving involves breaking down a problem into smaller, more manageable parts. It requires applying mathematical concepts like weighted averages, ratios, and logical thinking. This exercise is an excellent example of how to approach complex problems.

In problem solving:

• Start by identifying all the information given. For instance, costs of colors and their usage ratio were fundamental here.
• Next, decide on the appropriate mathematical tools to use, such as weighted averages when dealing with uneven contributions to an average like in this exercise.
• Perform calculations step-by-step, allowing for verification at each stage.
• Finally, interpret the result within the context of the problem, including checking if the result makes sense or matches any of the given options.

Effective problem solving in mathematics extends beyond finding a numerical answer; it includes understanding the solution process and the implications of your results.