Calculate the Price of Hydrogen in a Gas Mixture

Q: Why do we multiply percentages by individual prices?

Each gas contributes to the mixture's total cost based on both its percentage and its individual price. For example, helium is 3% of the mixture at $7 per 100g, so it contributes 0.03 × 7 = $0.21 to each 100g of mixture.

Q: What does the X represent in this problem?

X represents the selling price of the entire mixture per 100 grams. It's the total cost you'd pay for 100g of this gas mixture, which contains all three gases combined.

Q: Why do I need to divide by 0.87 at the end?

Because $ 0.87 \times H = X - 1.31 $ gives you hydrogen's contribution to the mixture cost. To find H (hydrogen's actual price per 100g), you must solve for H by dividing both sides by 0.87.

Q: How can I check if 1.15X - 1.51 is reasonable?

Substitute a test value! If X = $5, then hydrogen costs $ 1.15(5) - 1.51 = $4.24 $ per 100g. Since hydrogen makes up 87% of a $5 mixture, paying $4.24 for the hydrogen component makes sense!

Q: What if I get a negative price for hydrogen?

This means X is too small! Since $ H = 1.15X - 1.51 $, you need X > 1.31 for hydrogen to have a positive price. Check that your mixture price X is realistic given the other gas costs.

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

A mixture contains 3 gases:

Helium constitutes 3% of the mixture and costs $7 per 100 grams.

Hydrogen constitutes 87% of the mixture.

Oxygen constitutes 10% of the mixture and costs $11 per 100 grams.

If the mixture sells for $X per 100 grams, then what is the price of hydrogen?

2

Step-by-step solution

To solve this problem, we'll apply a weighted average approach to determine the cost of hydrogen in the mixture:

Step 1: Calculate the contribution of helium per 100 grams. $\text{Cost of helium per 100 grams} = 3\% \times 7 = 0.21$ dollars.
Step 2: Calculate the contribution of oxygen per 100 grams. $\text{Cost of oxygen per 100 grams} = 10\% \times 11 = 1.1$ dollars.
Step 3: Write the equation for the total cost per 100 grams. $X = 0.21 + \text{Cost of hydrogen per 100 grams} + 1.1$
Step 4: Simplify the equation to solve for the cost of hydrogen per 100 grams. $\text{Cost of hydrogen per 100 grams} = X - (0.21 + 1.1)$ $\text{Cost of hydrogen per 100 grams} = X - 1.31$
Step 5: Express the cost of hydrogen based on its percentage in the mixture. Since hydrogen makes up 87%, $\frac{87}{100} \times \text{Cost of hydrogen per 100 grams} = X - 1.31$
Step 6: Solve for the cost of hydrogen. $\text{Cost of hydrogen} = \frac{100}{87} \times (X - 1.31) \approx 1.15X - 1.51$

Therefore, the price of hydrogen is $1.15X - 1.51$ dollars per 100 grams.

3

Final Answer

$1.15x-1.51$ $

Key Points to Remember

Essential concepts to master this topic

Weighted Average Rule: Total cost equals sum of each component's percentage times its price
Technique: Set up equation X = 0.03(7) + 0.87(H) + 0.10(11) where H is hydrogen price
Check: Verify 1.15X - 1.51 gives reasonable hydrogen cost when X > 1.31 ✓

Common Mistakes

Avoid these frequent errors

Confusing percentage contribution with actual price per 100 grams
Don't solve 87% × H = X - 1.31 thinking H is the hydrogen price per 100g = wrong units! The 87% tells you hydrogen's weight fraction, not its cost contribution. Always recognize that H represents hydrogen's price per 100g, so 87% × H gives hydrogen's cost contribution to the mixture.

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 we multiply percentages by individual prices?

+

Each gas contributes to the mixture's total cost based on both its percentage and its individual price. For example, helium is 3% of the mixture at $7 per 100g, so it contributes 0.03 × 7 = $0.21 to each 100g of mixture.

What does the X represent in this problem?

+

X represents the selling price of the entire mixture per 100 grams. It's the total cost you'd pay for 100g of this gas mixture, which contains all three gases combined.

Why do I need to divide by 0.87 at the end?

+

Because $0.87 \times H = X - 1.31$ gives you hydrogen's contribution to the mixture cost. To find H (hydrogen's actual price per 100g), you must solve for H by dividing both sides by 0.87.

How can I check if 1.15X - 1.51 is reasonable?

+

Substitute a test value! If X = $5, then hydrogen costs $1.15(5) - 1.51 = $4.24$ per 100g. Since hydrogen makes up 87% of a $5 mixture, paying $4.24 for the hydrogen component makes sense!

What if I get a negative price for hydrogen?

+

This means X is too small! Since $H = 1.15X - 1.51$ , you need X > 1.31 for hydrogen to have a positive price. Check that your mixture price X is realistic given the other gas costs.

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