Solve the Ratio Problem: Finding Distribution of 80 Items in 3:5:2 Proportion

Q: Why do I add the ratio parts together first?

The ratio parts (3, 5, 2) represent relative sizes, not actual amounts. Adding them gives you the total number of equal units that make up your whole amount of 80 pencils.

Q: Can I solve this by setting up equations instead?

Yes! You could write $ 3x + 5x + 2x = 80 $ where x is the unit value. Solving gives $ x = 8 $, then multiply: 3×8, 5×8, 2×8.

Q: How do I know which pencil case gets which amount?

The order matters! The ratio 3:5:2 means the first case gets 3 parts, second gets 5 parts, third gets 2 parts. Always match the position in the ratio to the pencil case.

Ratio Distribution with Three-Part Proportions

The ratio between pencils in the pencil cases is 3:5:2.
If given that the total number of pencils is 80,
how many pencils are in each pencil case?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The ratio between pencils in the pencil cases is 3:5:2.
If given that the total number of pencils is 80,
how many pencils are in each pencil case?

Step-by-step solution

The solution involves distributing 80 pencils across three cases following a ratio of 3:5:2.

To find the number of pencils in each pencil case, follow these steps:

Step 1: Calculate the sum of the ratio parts. Here, the sum is $3 + 5 + 2 = 10$ .
Step 2: Calculate the number of pencils per unit of ratio: - Each unit is $\frac{80}{10} = 8$ pencils.
Step 3: Calculate the number of pencils in each case using the ratio:
Pencil case A: $3 \times 8 = 24$ ,
Pencil case B: $5 \times 8 = 40$ ,
Pencil case C: $2 \times 8 = 16$ .

Thus, the number of pencils in each case is:

Pencil case A 24 pencils, Pencil case B 40 pencils, Pencil case C 16 pencils

Final Answer

Pencil case A 24 pencils,
Pencil case B 40 pencils,
Pencil case C 16 pencils

Key Points to Remember

Essential concepts to master this topic

Sum Rule: Add all ratio parts to find total units
Unit Value: Divide total by sum: $\frac{80}{10} = 8$ pencils per unit
Verification: Check that 24 + 40 + 16 = 80 total pencils ✓

Common Mistakes

Avoid these frequent errors

Dividing 80 by each ratio part separately
Don't divide 80 by 3, 5, and 2 separately = gives impossible answers like 26.7, 16, and 40! This ignores that ratios are proportional parts of the whole. Always add ratio parts first (3+5+2=10), then divide total by this sum.

Practice Quiz

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What is the ratio between the orange and gray parts in the drawing?

FAQ

Everything you need to know about this question

Why do I add the ratio parts together first?

The ratio parts (3, 5, 2) represent relative sizes, not actual amounts. Adding them gives you the total number of equal units that make up your whole amount of 80 pencils.

What if the ratio parts don't add up evenly?

If your calculation gives decimal answers, double-check your arithmetic! Ratio problems with whole number totals should give whole number answers for each part.

Can I solve this by setting up equations instead?

Yes! You could write $3x + 5x + 2x = 80$ where x is the unit value. Solving gives $x = 8$ , then multiply: 3×8, 5×8, 2×8.

How do I know which pencil case gets which amount?

The order matters! The ratio 3:5:2 means the first case gets 3 parts, second gets 5 parts, third gets 2 parts. Always match the position in the ratio to the pencil case.

What if I get the ratio in a different order like 5:3:2?

Follow the same process! The method doesn't change - just make sure you assign the correct amounts to the right pencil cases based on their position in the given ratio.

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