Samit was given some money to take care of his travel during a 6-day sales drive he had to undertake. However, he had to increase his stay by another 4 days and as a result his average daily travel allowance went down by Rs. 56. What was the amount that was sanctioned to him in the beginning?
Rs. 840
Understanding the Travel Allowance Problem
This problem involves calculating an initial travel allowance based on how the average daily amount changes when the duration of a trip is extended. We are given the initial duration, the extended duration, and the decrease in the average daily allowance.
Let's identify the key information:
- Initial number of days: 6
- Additional days: 4
- Extended number of days: \(6 + 4 = 10\)
- Decrease in average daily allowance: Rs. 56
- We need to find the initial total amount sanctioned.
Setting Up the Equation for Daily Average Allowance
Let \(A\) be the total amount of money sanctioned to Samit in the beginning. This amount remained the same for the extended trip as well.
- The initial average daily travel allowance was the total amount divided by the initial number of days: \( \text{Initial Average} = \frac{A}{6} \)
- When the stay was extended, the total amount \(A\) was spread over the new number of days (10 days). The new average daily travel allowance is: \( \text{Extended Average} = \frac{A}{10} \)
The problem states that the average daily travel allowance went down by Rs. 56 as a result of the extended stay. This means the initial average was Rs. 56 more than the extended average.
We can write this relationship as an equation:
\[ \text{Initial Average} - \text{Extended Average} = 56 \] \[ \frac{A}{6} - \frac{A}{10} = 56 \]Solving for the Initial Sanctioned Amount
Now we need to solve the equation \( \frac{A}{6} - \frac{A}{10} = 56 \) for \(A\).
To solve for \(A\), we first need to find a common denominator for the fractions. The least common multiple (LCM) of 6 and 10 is 30.
Multiply both sides of the equation by 30 to eliminate the denominators:
\[ 30 \left( \frac{A}{6} - \frac{A}{10} \right) = 30 \times 56 \] \[ 30 \times \frac{A}{6} - 30 \times \frac{A}{10} = 1680 \] \[ 5A - 3A = 1680 \]Combine the terms with \(A\):
\[ 2A = 1680 \]Now, isolate \(A\) by dividing both sides by 2:
\[ A = \frac{1680}{2} \] \[ A = 840 \]So, the amount that was sanctioned to him in the beginning was Rs. 840.
Verifying the Solution
Let's check if our calculated amount (Rs. 840) satisfies the conditions given in the problem.
- Initial amount \(A = 840\).
- Initial duration = 6 days.
- Initial average daily allowance \( = \frac{840}{6} = 140 \) Rs.
- Extended duration = 10 days.
- Extended average daily allowance \( = \frac{840}{10} = 84 \) Rs.
The difference between the initial average and the extended average is \(140 - 84 = 56\) Rs.
This matches the information given in the problem (the average daily travel allowance went down by Rs. 56). Therefore, our calculated amount of Rs. 840 is correct.
| Scenario | Total Amount | Duration (Days) | Average Daily Allowance |
|---|---|---|---|
| Initial Plan | Rs. 840 | 6 | Rs. 140 |
| Extended Stay | Rs. 840 | 10 | Rs. 84 |
| Difference in Average | Rs. \(140 - 84 = 56\) |
Revision Table: Key Concepts in Word Problems
| Concept | Explanation | How it Applies Here |
|---|---|---|
| Identifying Variables | Represent unknown quantities with letters (variables). | We used \(A\) for the initial sanctioned amount. |
| Formulating Equations | Translate the word problem's relationships into mathematical equations. | We set up the equation \( \frac{A}{6} - \frac{A}{10} = 56 \). |
| Solving Equations | Use algebraic techniques to find the value of the variable. | We solved for \(A\) by finding a common denominator and simplifying. |
| Verification | Check if the solution satisfies the original conditions of the problem. | We calculated the initial and extended averages and confirmed the difference was Rs. 56. |
Additional Information: Average Calculations
The average is a fundamental concept used in many areas, including finances, statistics, and daily life. It is calculated by dividing the total sum of values by the number of values.
Formula for Average:
\[ \text{Average} = \frac{\text{Total Sum}}{\text{Number of Items}} \]In this problem, the "Total Sum" is the total travel allowance, and the "Number of Items" is the number of days the trip lasted.
Understanding how averages change when the total or the number of items changes is important. In this case, the total amount remained constant, but the number of days increased, which naturally caused the average daily allowance to decrease.
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