Vishnu spends Rs. 5000 in buying 12 tables and some chairs. The cost of one table is Rs. 50 and that of one chair is Rs. 40. What is the ratio of the numbers of the chairs to the number of tables purchased?
55 : 6
Calculating the Ratio of Chairs to Tables Purchased
The problem asks us to find the ratio of the number of chairs to the number of tables purchased by Vishnu, given the total expenditure, the number of tables, and the cost per table and per chair.
Let's break down the information provided:
- Total amount spent by Vishnu: Rs. 5000
- Number of tables purchased: 12
- Cost of one table: Rs. 50
- Cost of one chair: Rs. 40
Step-by-Step Calculation
First, we need to calculate the total cost of the tables purchased.
- Cost of 1 table = Rs. 50
- Number of tables = 12
- Total cost of tables = Number of tables $\times$ Cost per table
- Total cost of tables = $12 \times 50 = 600$
So, Vishnu spent Rs. 600 on tables.
Next, we find the amount of money spent on chairs. This is the total expenditure minus the cost of tables.
- Amount spent on chairs = Total amount spent - Total cost of tables
- Amount spent on chairs = $5000 - 600 = 4400$
Vishnu spent Rs. 4400 on chairs.
Now, we can calculate the number of chairs purchased using the amount spent on chairs and the cost per chair.
- Cost of one chair = Rs. 40
- Amount spent on chairs = Rs. 4400
- Number of chairs = Amount spent on chairs $/$ Cost per chair
- Number of chairs = $4400 / 40 = 110$
Vishnu purchased 110 chairs.
Finally, we need to find the ratio of the number of chairs to the number of tables purchased.
- Number of chairs = 110
- Number of tables = 12
- Ratio of chairs to tables = Number of chairs : Number of tables
- Ratio = $110 : 12$
To simplify the ratio, we find the greatest common divisor (GCD) of 110 and 12, which is 2. We divide both numbers by 2.
- $110 \div 2 = 55$
- $12 \div 2 = 6$
The simplified ratio of the number of chairs to the number of tables is $55 : 6$.
Let's summarize the key values:
| Item | Number Purchased | Cost Per Item (Rs.) | Total Cost (Rs.) |
|---|---|---|---|
| Tables | 12 | 50 | $12 \times 50 = 600$ |
| Chairs | Calculated (110) | 40 | $5000 - 600 = 4400$ |
The number of chairs purchased is 110, and the number of tables purchased is 12. The ratio of chairs to tables is 110 : 12, which simplifies to 55 : 6.
Revision Table: Furniture Ratio Calculation
| Concept | Description | Calculation Used |
|---|---|---|
| Total Cost of Items | Sum of costs of all purchased items. | Cost of Tables + Cost of Chairs |
| Cost of Multiple Items | Number of items multiplied by the cost of one item. | Number $\times$ Unit Cost |
| Number of Items from Cost | Total cost for an item type divided by the unit cost. | Total Cost $/$ Unit Cost |
| Ratio | Comparison of two quantities by division. | Quantity 1 : Quantity 2 (or $\frac{\text{Quantity 1}}{\text{Quantity 2}}$) |
| Simplifying Ratios | Dividing both parts of the ratio by their greatest common divisor. | $a : b = (a/GCD) : (b/GCD)$ |
Additional Information: Understanding Ratios
A ratio is a way to compare two or more quantities. It shows how many times one value contains or is contained within the other. Ratios can be written in a few ways:
- Using a colon (e.g., $a : b$)
- As a fraction (e.g., $\frac{a}{b}$)
- Using the word "to" (e.g., $a$ to $b$)
In this problem, we compared the number of chairs to the number of tables. The order matters in a ratio. The ratio of chairs to tables ($110 : 12$ or $55 : 6$) is different from the ratio of tables to chairs ($12 : 110$ or $6 : 55$). Always ensure you are presenting the ratio in the order requested by the question.
Simplifying ratios makes them easier to understand and compare. It's similar to simplifying fractions. You divide both parts of the ratio by the largest number that divides into both evenly (their GCD).
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