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The 10th-grade class officers ordered t-shirts and sweatshirts to sell as a fundraiser. Each t-shirt weighed 1/2 of a pound, and each sweatshirt weighed 3/4 of a pound. The total weight of the contents of the box was 43 3/4 pounds.

Which equation written in standard form represents the number of t-shirts, [tex]\( t \)[/tex], and the number of sweatshirts, [tex]\( s \)[/tex], that were ordered?

A. [tex]\( \frac{1}{2}t + \frac{3}{4}s = 43 \frac{3}{4} \)[/tex]

B. [tex]\( \frac{3}{4}s = -\frac{1}{2}t + 43 \frac{3}{4} \)[/tex]

C. [tex]\( 2t + 3s = 175 \)[/tex]

D. [tex]\( 3s = -2t + 175 \)[/tex]



Answer :

GinnyAnswer
Sure, let's go through the steps together to determine the correct equations in standard form.

Initially, we are given the following equations based on the weights:

1. [tex]\(\frac{1}{2} t + \frac{3}{4} s = 43 \frac{3}{4}\)[/tex]
2. [tex]\(\frac{3}{4} s = -\frac{1}{2} t + 43 \frac{3}{4}\)[/tex]
3. [tex]\(2 t + 3 s = 175\)[/tex]
4. [tex]\(3 s = -2 t + 175\)[/tex]

### Step 1: Convert each equation to standard form (if necessary)

Equation 1:
[tex]\[\frac{1}{2} t + \frac{3}{4} s = 43 \frac{3}{4}\][/tex]
- Multiply through by 4 to clear the fractions:
[tex]\[4 \left(\frac{1}{2} t + \frac{3}{4} s\right) = 4 \times 43 \frac{3}{4}\][/tex]
[tex]\[2 t + 3 s = 175\][/tex]

Equation 2:
[tex]\[\frac{3}{4} s = -\frac{1}{2} t + 43 \frac{3}{4}\][/tex]
- Multiply through by 4 to clear the fractions:
[tex]\[4 \left(\frac{3}{4} s\right) = 4 \left(-\frac{1}{2} t + 43 \frac{3}{4}\right)\][/tex]
[tex]\[3 s = -2 t + 175\][/tex]

### Step 2: Verify the standard form of the equations

Equation 3:
[tex]\[2 t + 3 s = 175\][/tex]
- We notice that this is already provided in standard form.

Equation 4:
[tex]\[3 s = -2 t + 175\][/tex]
- Similarly, this is already in standard form.

### Conclusion:

After converting the given equations into standard forms, we see that:

[tex]\[ 2t + 3s = 175 \][/tex]
is a valid standard form equation representing the relationship between the number of T-shirts [tex]\( t \)[/tex] and sweatshirts [tex]\( s \)[/tex] ordered by the students.

Therefore, the equation [tex]\( 2 t + 3 s = 175 \)[/tex] represents the number of T-shirts [tex]\( t \)[/tex] and the number of sweatshirts [tex]\( s \)[/tex] that were ordered.

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