Let's solve this step by step.

Step 1: Define the variables

• Let $x$ be the student's initial average before the Banking test.
• Let $n$ be the number of tests taken before the Banking test.

Step 2: Calculate the new average after the Banking test

After scoring 83 in the Banking test, the average increases by 2: $\text{New average after Banking test} = x + 2$ The total number of tests now is $n + 1$.

The equation for the new average is: $\frac{nx + 83}{n+1} = x + 2$

Step 3: Solve the equation for $n$ and $x$

Multiply both sides by $n + 1$ to get rid of the fraction: $nx + 83 = (x + 2)(n + 1)$ Expanding the right-hand side: $nx + 83 = nx + n + 2x + 2$ Cancel out $nx$ from both sides: $83 = n + 2x + 2$ Simplifying further: $n + 2x = 81 \quad \text{(Equation 1)}$

Step 4: Calculate the new average after the Accounting test

After scoring 75 in Accounting, the average increases by 1: $\text{New average after Accounting test} = x + 3$ The total number of tests is now $n + 2$.

The equation for the new average is: $\frac{nx + 83 + 75}{n+2} = x + 3$ Simplify the equation: $\frac{nx + 158}{n+2} = x + 3$

Multiply both sides by $n + 2$: $nx + 158 = (x + 3)(n + 2)$ Expanding the right-hand side: $nx + 158 = nx + 3n + 2x + 6$ Cancel out $nx$ from both sides: $158 = 3n + 2x + 6$ Simplify further: $3n + 2x = 152 \quad \text{(Equation 2)}$

Step 5: Solve the system of equations

We now have two equations:

1. $n + 2x = 81$
2. $3n + 2x = 152$

Subtract Equation 1 from Equation 2: $(3n + 2x) - (n + 2x) = 152 - 81$ Simplifying: $2n = 71$ $n = 35.5$ Substitute $n = 35.5$ into Equation 1: $35.5 + 2x = 81$ $2x = 45.5$ $x = 22.75$

Step 6: Calculate the average after the Reasoning test

The student will have taken $n + 3 = 38.5$ tests, and the total marks will be: $\text{Total marks} = nx + 83 + 75 + 51$ $\text{New average} = \frac{35.5 \times 22.75 + 83 + 75 + 51}{38.5}$ Calculate: $\text{Total marks} = 35.5 \times 22.75 + 83 + 75 + 51 = 808.125 + 209 = 1017.125$ $\text{New average} = \frac{1017.125}{38.5} \approx 26.41$

Thus, the student's new average after the Reasoning test would be approximately 26.41.

Would you like further details or clarifications? Here are some related questions:

1. How do different types of averages (mean, median, mode) apply to test scores?
2. Can this method be used to calculate cumulative GPA?
3. How does the number of tests affect the impact of each new score on the average?
4. What is the difference between weighted and unweighted averages?
5. How can averages be misleading in different contexts?

Tip: Always double-check calculations, especially when working with averages and cumulative data, to ensure accuracy.