Potatoes And Grapes: Inequality For Total Cost < $5

by ADMIN 52 views

Alright, guys, let's break down this math problem! Dan's at the store, trying to buy some potatoes and grapes, but he's on a budget. We need to figure out which inequality best describes his shopping situation. Here’s the lowdown:

  • Potatoes: $x$ pounds at $0.85$ per pound
  • Grapes: $y$ pounds at $1.29$ per pound
  • Total cost: Less than $5$

Setting Up the Inequality

The main goal here is to translate the word problem into a mathematical inequality. Remember, an inequality is just like an equation, but instead of an equals sign (=), we use symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

First, let's figure out how to represent the total cost of the potatoes. If each pound costs $0.85$, then the total cost for $x$ pounds is simply $0.85x$. Easy peasy!

Next, let's do the same for the grapes. If each pound costs $1.29$, then the total cost for $y$ pounds is $1.29y$. Got it?

Now, we need to add those two costs together to get the total cost of Dan's purchase: $0.85x + 1.29y$.

Finally, the problem tells us that the total cost was less than $5$. So, we use the "less than" symbol (<) to represent this: $0.85x + 1.29y < 5$.

Why This Inequality Makes Sense

Let's think about why this inequality is the right one. Imagine Dan buys 1 pound of potatoes and 1 pound of grapes. The total cost would be:

0.85(1)+1.29(1)=0.85+1.29=2.140.85(1) + 1.29(1) = 0.85 + 1.29 = 2.14

Since $2.14$ is less than $5$, this combination of potatoes and grapes fits our inequality. If Dan bought a lot more potatoes and grapes, the total cost would go up, and at some point, it would exceed $5$. The inequality helps us define all the possible combinations of potatoes and grapes that Dan can buy without spending more than $5$.

Common Mistakes to Avoid

  • Using the wrong symbol: Make sure you understand the difference between < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). In this case, "less than" is crucial.
  • Incorrectly setting up the equation: Double-check that you're multiplying the price per pound by the correct number of pounds for both potatoes and grapes.
  • Forgetting to include both variables: The inequality must include both $x$ (pounds of potatoes) and $y$ (pounds of grapes) to accurately represent the total cost.

Practical Implications

This type of problem is super useful in real life! Whenever you're budgeting or trying to figure out how much of different items you can buy with a limited amount of money, you're essentially using inequalities. Whether you're buying groceries, school supplies, or even planning a party, understanding inequalities can help you make smart financial decisions.

Let's Recap

To solve this problem, we:

  1. Identified the variables: $x$ (pounds of potatoes) and $y$ (pounds of grapes).
  2. Determined the cost of each item: $0.85x$ for potatoes and $1.29y$ for grapes.
  3. Combined the costs and set up the inequality: $0.85x + 1.29y < 5$.

So, there you have it! The inequality $0.85x + 1.29y < 5$ represents Dan's purchase of potatoes and grapes with a total cost less than $5$. Keep practicing these types of problems, and you'll become a master of inequalities in no time!

Real-World Application

Imagine you're planning a fruit salad for a party and have a budget of $20. You want to buy apples that cost $2 per pound and oranges that cost $1.50 per pound. Let 'a' be the number of pounds of apples and 'o' be the number of pounds of oranges. The inequality that represents this scenario is:

2a+1.50o≤202a + 1.50o \le 20

This inequality helps you determine the possible combinations of apples and oranges you can buy without exceeding your $20 budget. For example, you could buy 5 pounds of apples and 6 pounds of oranges:

2(5)+1.50(6)=10+9=192(5) + 1.50(6) = 10 + 9 = 19

Since $19 is less than or equal to $20, this combination works. However, if you tried to buy 8 pounds of apples and 5 pounds of oranges:

2(8)+1.50(5)=16+7.50=23.502(8) + 1.50(5) = 16 + 7.50 = 23.50

This exceeds your budget, so it's not a valid option. Understanding and using inequalities allows you to make informed decisions when planning purchases and staying within budget.

Another Example: Snack Mix

Let's say you're making a snack mix with pretzels and nuts. Pretzels cost $3 per pound, and nuts cost $5 per pound. You want to spend no more than $15. If 'p' represents the pounds of pretzels and 'n' represents the pounds of nuts, the inequality is:

3p+5n≤153p + 5n \le 15

This inequality shows all the possible amounts of pretzels and nuts you can buy without going over your $15 budget. You could buy 2 pounds of pretzels and 1 pound of nuts:

3(2)+5(1)=6+5=113(2) + 5(1) = 6 + 5 = 11

Since $11 is less than or equal to $15, this works. But if you tried to buy 3 pounds of pretzels and 2 pounds of nuts:

3(3)+5(2)=9+10=193(3) + 5(2) = 9 + 10 = 19

This is over your budget, so it's not an option. Inequalities are very helpful in managing costs and making choices that fit within your financial limits.

Conclusion

Inequalities are a fundamental tool in mathematics and have many practical applications in everyday life. Understanding how to set up and solve inequalities can help you make informed decisions about budgeting, planning, and resource allocation. Whether you're shopping for groceries, planning a party, or managing a project, inequalities provide a framework for staying within your limits and optimizing your resources. Keep practicing with different scenarios to build your skills and confidence in using inequalities effectively. Remember, math isn't just about numbers; it's about making smart choices and solving real-world problems!