Solving Olympiad Exam Problem: Finding Total Students

This problem involves calculating the total number of students (boys and girls) who appeared for an Olympiad exam, given information about the percentages of students who failed and relationships between the numbers of students who passed.

Defining Variables

Let's represent the unknown quantities using variables:

• Let \(B\) be the total number of boys who appeared for the exam.
• Let \(G\) be the total number of girls who appeared for the exam.

The total number of students who appeared is \(B + G\).

Calculating Failed and Passed Students by Gender

We are given the failure percentages for boys and girls:

• 20% of boys failed. So, the number of boys who failed is \(0.20 \times B\).
• 15% of girls failed. So, the number of girls who failed is \(0.15 \times G\).

If 20% of boys failed, then the percentage of boys who passed is \(100\% - 20\% = 80\%\). The number of boys who passed is \(0.80 \times B\).

If 15% of girls failed, then the percentage of girls who passed is \(100\% - 15\% = 85\%\). The number of girls who passed is \(0.85 \times G\).

Setting Up Equations from Given Information

We are given two key pieces of information that we can translate into equations:

1. A total of 90 students failed.
2. The number of boys who passed was 70 more than that of the girls who passed.

From point 1:

Number of boys who failed + Number of girls who failed = Total failed students

\(0.20 B + 0.15 G = 90\) (Equation 1)

From point 2:

Number of boys who passed = Number of girls who passed + 70

\(0.80 B = 0.85 G + 70\)

Rearranging this equation, we get:

\(0.80 B - 0.85 G = 70\) (Equation 2)

Solving the System of Equations

Now we have a system of two linear equations with two variables, B and G:

1) \(0.20 B + 0.15 G = 90\)

2) \(0.80 B - 0.85 G = 70\)

We can solve this system using methods like substitution or elimination. Let's use elimination. We can multiply Equation 1 by 4 to make the coefficient of B equal to that in Equation 2:

\(4 \times (0.20 B + 0.15 G) = 4 \times 90\)

\(0.80 B + 0.60 G = 360\) (Equation 3)

Now, subtract Equation 2 from Equation 3:

\((0.80 B + 0.60 G) - (0.80 B - 0.85 G) = 360 - 70\)

\(0.80 B + 0.60 G - 0.80 B + 0.85 G = 290\)

\((0.60 + 0.85) G = 290\)

\(1.45 G = 290\)

Now, solve for G:

\(G = \frac{290}{1.45}\)

\(G = \frac{29000}{145}\)

\(G = 200\)

So, the number of girls who appeared is 200.

Now substitute the value of G (200) back into Equation 1 to find B:

\(0.20 B + 0.15 (200) = 90\)

\(0.20 B + 30 = 90\)

\(0.20 B = 90 - 30\)

\(0.20 B = 60\)

Now, solve for B:

\(B = \frac{60}{0.20}\)

\(B = \frac{6000}{20}\)

\(B = 300\)

So, the number of boys who appeared is 300.

Calculating the Total Number of Students

The total number of students who appeared for the exam is the sum of the number of boys and the number of girls.

Total students = \(B + G = 300 + 200 = 500\)

Verification

Let's quickly check if these numbers satisfy the original conditions:

• Boys failed: 20% of 300 = \(0.20 \times 300 = 60\)
• Girls failed: 15% of 200 = \(0.15 \times 200 = 30\)
• Total failed: \(60 + 30 = 90\) (Matches)
• Boys passed: 80% of 300 = \(0.80 \times 300 = 240\)
• Girls passed: 85% of 200 = \(0.85 \times 200 = 170\)
• Difference in passed: \(240 - 170 = 70\) (Matches)

The numbers satisfy all the given conditions.

Conclusion on Olympiad Exam Students

The total number of students that appeared for the Olympiad exam was 500.

Revision Table: Key Concepts

Additional Information: Percentage Problems

Percentage problems often require converting percentages to decimals or fractions to perform calculations. Remember that "percent" means "out of one hundred."

• To convert a percentage to a decimal, divide by 100. For example, 20% = \(20/100 = 0.20\).
• To calculate a percentage of a number, multiply the decimal or fraction form of the percentage by the number. For example, 20% of 300 is \(0.20 \times 300\).

Word problems like this one test your ability to understand the relationships described and translate them into mathematical expressions. Carefully define your variables and set up your equations based on the given conditions.