Calculate Javier's Weighted Exam Score: Solve for x when 0.55x + 0.45(76) = 84

Q: Why do I need to convert percentages to decimals?

Percentages represent parts out of 100, but for calculations you need the actual decimal value. For example, 55% means 55/100 = 0.55. Using 55 instead of 0.55 makes your answer 100 times too big!

Q: How do I set up a weighted average equation?

Use the formula: Final Average = (Weight₁ × Score₁) + (Weight₂ × Score₂). Make sure your weights add up to 1.0 (like 0.55 + 0.45 = 1.0).

Q: Can I solve this without converting to decimals?

Yes! You could write $ 84 = \frac{55}{100}x + \frac{45}{100}(76) $, but converting to decimals first makes the arithmetic much easier and reduces calculation errors.

Q: How do I check if my weighted average is correct?

Substitute your answer back: $ 0.55(90.55) + 0.45(76) = 49.8025 + 34.2 = 84 $. The result should equal the given final average!

Step-by-step video solution

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Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

How much does Javier score on the first exam that has a weight of 55%, given that he scores 76 on the second exam and his final average is 84?

2

Step-by-step solution

To solve this problem, we'll proceed with the following steps:

Step 1: Define the weights and scores for each exam.
Step 2: Formulate the equation for the weighted average.
Step 3: Solve the equation for $x$ , the unknown score on the first exam.

Now, let's work through each step:

Step 1: We know:

Weight of the first exam: $0.55$
Weight of the second exam: $0.45$
Score on the second exam: $76$
Final average score: $84$
Unknown score (first exam): Let it be $x$

Step 2: We use the weighted average formula:

84 = 0.55 \times x + 0.45 \times 76

Step 3: Now solve for $x$ .

First, calculate the contribution of the second exam:

0.45 \times 76 = 34.2

Substitute this back into the equation:

84 = 0.55 \times x + 34.2

Subtract 34.2 from both sides to solve for $0.55 \times x$ :

84 - 34.2 = 0.55 \times x

49.8 = 0.55 \times x

Finally, divide both sides by 0.55 to isolate $x$ :

x = \frac{49.8}{0.55}

Calculate the result:

x \approx 90.55

Therefore, the solution to the problem is $\mathbf{90.55}$ .

3

Final Answer

$90.55$

Key Points to Remember

Essential concepts to master this topic

Weighted Average Formula: Final average equals sum of weighted individual scores
Technique: Convert percentages to decimals: 55% becomes 0.55 × x
Check: Substitute x = 90.55: 0.55(90.55) + 0.45(76) = 84 ✓

Common Mistakes

Avoid these frequent errors

Using percentages instead of decimals in calculations
Don't use 55% and 45% directly in equations = wrong scale! This gives answers 100 times larger than correct. Always convert percentages to decimals first: 55% = 0.55, 45% = 0.45.

Practice Quiz

Test your knowledge with interactive questions

A hotel's overall rating is determined according to a weighted average of several categories. Each category is given a rating and a weighted factor. Below are the ratings for the "Happy Tourist" hotel:

Determine the hotel's overall rating?

FAQ

Everything you need to know about this question

Why do I need to convert percentages to decimals?

+

Percentages represent parts out of 100, but for calculations you need the actual decimal value. For example, 55% means 55/100 = 0.55. Using 55 instead of 0.55 makes your answer 100 times too big!

How do I set up a weighted average equation?

+

Use the formula: Final Average = (Weight₁ × Score₁) + (Weight₂ × Score₂). Make sure your weights add up to 1.0 (like 0.55 + 0.45 = 1.0).

What if the weights don't add up to 100%?

+

Real weighted averages should always total 100% (or 1.0 in decimal form). If they don't, double-check the problem - there might be missing information or an error in the given weights.

Can I solve this without converting to decimals?

+

Yes! You could write $84 = \frac{55}{100}x + \frac{45}{100}(76)$ , but converting to decimals first makes the arithmetic much easier and reduces calculation errors.

How do I check if my weighted average is correct?

+

Substitute your answer back: $0.55(90.55) + 0.45(76) = 49.8025 + 34.2 = 84$ . The result should equal the given final average!

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Calculate Javier's Weighted Exam Score: Solve for x when 0.55x + 0.45(76) = 84

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