Calculating Marbles in Bags: Determining the 9 Count from an Average of 9.64

Q: Why can't I just divide the remaining marbles by 9?

Because we don't know the total number of marbles yet! The average of 9.64 applies to ALL bags combined, so we need to set up an equation that includes both known and unknown bags.

Q: What does the variable n represent exactly?

The variable n represents the number of bags containing exactly 9 marbles each. This is what we're trying to find in the problem.

Q: Why did we get 8.0625 but round to 8?

Because we can't have a fractional number of bags! The problem asks for whole bags, so we need a whole number answer. The small rounding difference is due to the given average being approximate.

Q: How can I verify my answer is correct?

Substitute n = 8 back into the average formula: $ \frac{63 + 9(8)}{6 + 8} = \frac{135}{14} = 9.64 $. Since this matches the given average, our answer is correct!

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

Each bag of marbles contains an average of $9.64$ marbles.

The first bag has 18 marbles, another two have 12 marbles, and the last three have 7 marbles.

How many bags contain 9 marbles?

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the number of bags and compute the total number of marbles.
Step 2: Apply the formula for the average and solve for the unknown.
Step 3: Solve the equation to determine the number of bags containing 9 marbles.

Let's work through these steps:

Step 1: The problem details that the average number of marbles per bag is 9.64. We need to find the total number of bags. Let $n$ be the number of bags that contain 9 marbles. Calculating the total number of bags is necessary as a starting point.

Given bags and their marbles:
- 1 bag containing 18 marbles
- 2 bags containing 12 marbles each: Total $= 2 \times 12 = 24$ marbles
- 3 bags containing 7 marbles each: Total $= 3 \times 7 = 21$ marbles

Total marbles in known bags = $18 + 24 + 21 = 63$ marbles

Step 2: Use the total average calculation to find the number of bags:

\text{Number of unknown bags} = n

Total marbles = $63 + 9n$

The number of total bags is $1 + 2 + 3 + n = 6 + n$ .

Thus, by the average formula:

\frac{63 + 9n}{6 + n} = 9.64

Step 3: Solve for $n$ .

63 + 9n = 9.64(6 + n)

63 + 9n = 57.84 + 9.64n

Subtract $57.84$ from both sides:

63 - 57.84 = 9.64n - 9n

5.16 = 0.64n

Divide by 0.64 to solve for $n$ :

n = \frac{5.16}{0.64} = 8.0625

This result implies rounding is needed, thus $n = 8$ bags.

Therefore, the solution to the problem is $8$ bags.

3

Final Answer

$8$ bags

Key Points to Remember

Essential concepts to master this topic

Formula: Average equals total marbles divided by total bags
Setup: Let n = unknown bags, then $\frac{63 + 9n}{6 + n} = 9.64$
Check: Substitute n = 8: $\frac{135}{14} = 9.64$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to include unknown bags in total bag count
Don't use only the 6 known bags in the denominator = wrong average calculation! This ignores the n bags with 9 marbles each and gives an incorrect equation. Always include ALL bags in your count: 6 + n total bags.

Practice Quiz

Test your knowledge with interactive questions

Norbert buys some new clothes.

When he gets home, he decides to work out how much each outfit cost him on average.

What answer should he come up with?

FAQ

Everything you need to know about this question

Why can't I just divide the remaining marbles by 9?

+

Because we don't know the total number of marbles yet! The average of 9.64 applies to ALL bags combined, so we need to set up an equation that includes both known and unknown bags.

What does the variable n represent exactly?

+

The variable n represents the number of bags containing exactly 9 marbles each. This is what we're trying to find in the problem.

How do I handle the decimal 9.64 in the equation?

+

Treat it like any other number! When you multiply both sides by $(6 + n)$ , you get $63 + 9n = 9.64(6 + n)$ . Then distribute the 9.64 to both terms inside the parentheses.

Why did we get 8.0625 but round to 8?

+

Because we can't have a fractional number of bags! The problem asks for whole bags, so we need a whole number answer. The small rounding difference is due to the given average being approximate.

How can I verify my answer is correct?

+

Substitute n = 8 back into the average formula: $\frac{63 + 9(8)}{6 + 8} = \frac{135}{14} = 9.64$ . Since this matches the given average, our answer is correct!

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