Banana Cost Equation: Finding The Price For 4 1/4 Pounds

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Hey guys! Let's dive into a fun math problem about bananas! This is a classic example of direct variation, where the cost of something changes proportionally with its weight. In this case, we're looking at how the cost of bananas relates to how much they weigh. We'll break down the steps to write an equation that shows this relationship and then use it to figure out the cost of a specific amount of bananas. Let’s get started!

Understanding Direct Variation with Bananas

When we say the cost of bananas varies directly with their weight, it means there's a constant ratio between the two. Think of it this way: if you double the weight, you double the cost. If you triple the weight, you triple the cost, and so on. This constant ratio is super important because it helps us create our equation. In mathematical terms, this relationship can be expressed as y = kx, where:

  • y is the total cost
  • x is the weight (in pounds)
  • k is the constant of variation (the price per pound)

Our first step is to find this k, the constant of variation, which will tell us the price per pound of bananas. The problem gives us some information to work with: Miguel bought 3 1/2 pounds of bananas for $1.12. We can use this data to calculate k. Once we know k, we can write the equation that relates the cost and weight of the bananas. Then, we’ll use the equation to find out how much 4 1/4 pounds of bananas would cost. So, stick with me, and we'll solve this together!

Step 1: Finding the Constant of Variation (k)

Okay, so we know that Miguel paid $1.12 for 3 1/2 pounds of bananas. To find the constant of variation (k), which is essentially the price per pound, we need to divide the total cost by the weight. First, let’s convert the mixed number 3 1/2 into an improper fraction to make our calculations easier. 3 1/2 is the same as (3 * 2 + 1) / 2, which equals 7/2. Now we have:

  • Total cost (y) = $1.12
  • Weight (x) = 7/2 pounds

Using the formula y = kx, we can plug in these values and solve for k:

$1.12 = k * (7/2)

To isolate k, we need to divide both sides of the equation by 7/2. Remember, dividing by a fraction is the same as multiplying by its reciprocal. So, we'll multiply both sides by 2/7:

$1.12 * (2/7) = k

Now, let's do the math. $1.12 multiplied by 2 is $2.24. Then, we divide $2.24 by 7:

$2.24 / 7 = $0.32

So, k = $0.32. This means the price per pound of bananas is $0.32. Now we've got a crucial piece of information that we'll use to write our equation and solve the rest of the problem.

Step 2: Writing the Equation

Now that we've found the constant of variation (k), which is $0.32, we can write the equation that relates the cost of the bananas to their weight. Remember the direct variation formula: y = kx. We simply plug in our value for k:

y = 0.32x

This equation tells us that the total cost (y) is equal to $0.32 times the weight in pounds (x). So, for every pound of bananas, the cost increases by $0.32. This equation is super handy because we can use it to find the cost of any weight of bananas. If you want to know the cost of 5 pounds, you just plug in 5 for x. If you want to know the cost of 10 pounds, you plug in 10 for x, and so on. Now, let's use this equation to find the cost of 4 1/4 pounds of bananas.

Step 3: Finding the Cost of 4 1/4 Pounds of Bananas

Alright, we've got our equation: y = 0.32x. Now we need to find the cost of 4 1/4 pounds of bananas. So, we're going to substitute 4 1/4 for x in our equation. First, let’s convert 4 1/4 into an improper fraction. 4 1/4 is the same as (4 * 4 + 1) / 4, which equals 17/4. Now we can plug this into our equation:

y = 0.32 * (17/4)

To make the multiplication easier, let’s convert 0.32 into a fraction as well. 0. 32 is the same as 32/100. We can simplify this fraction by dividing both the numerator and denominator by 4, which gives us 8/25. So our equation now looks like this:

y = (8/25) * (17/4)

Now we can multiply the fractions. Multiply the numerators (8 * 17) and the denominators (25 * 4):

y = 136 / 100

This fraction can be simplified. Both 136 and 100 are divisible by 4. Dividing both by 4 gives us:

y = 34 / 25

Now, let's convert this improper fraction back into a decimal to get the cost in dollars. 34 divided by 25 is 1.36. So:

y = $1.36

Therefore, the cost of 4 1/4 pounds of bananas is $1.36. We did it!

Final Answer and Summary

So, after working through the problem, we found that the equation relating the cost of bananas to their weight is y = 0.32x, where y is the total cost and x is the weight in pounds. We also figured out that 4 1/4 pounds of bananas would cost $1.36.

Let's recap the steps we took to solve this problem:

  1. We understood the concept of direct variation and the equation y = kx.
  2. We used the given information (Miguel's purchase) to find the constant of variation, k.
  3. We wrote the equation relating cost and weight using the value of k.
  4. We substituted the given weight (4 1/4 pounds) into the equation to find the cost.

Understanding direct variation is super useful in many real-life situations, from figuring out costs to understanding relationships between different quantities. Keep practicing, and you'll become a pro at solving these types of problems! Keep up the great work, guys!