Question

1. match each story to an equation.

a) a stack of nested paper cups is 8 inches tall. the first cup is 4 inches tall and each of the rest of the cups in the stack adds \\(\frac{1}{4}\\) inch to the height of the stack.
b) a baker uses 4 cups of flour. she uses \\(\frac{1}{4}\\) cup to flour the counters and the rest to make 8 identical muffins.
c) elena has an 8 - foot piece of ribbon. she cuts off a piece that is \\(\frac{1}{4}\\) of a foot long and cuts the remainder into four pieces of equal length.

1. \\(\frac{1}{4}+4x = 8\\)
2. \\(4 + \frac{1}{4}x = 8\\)
3. \\(8x + \frac{1}{4} = 4\\)

Explanation:

Part (a)

Step 1: Define variables

Let \( x \) be the number of additional cups (excluding the first cup). The height of the first cup is 4 inches, and each additional cup adds \( \frac{1}{4} \) inch. The total height is 8 inches.

Step 2: Form the equation

The total height is the height of the first cup plus the height added by the additional cups. So the equation is \( 4+\frac{1}{4}x = 8 \), which matches equation 2.

Part (b)

Step 1: Define variables

Let \( x \) be the amount of flour used per muffin. The baker uses \( \frac{1}{4} \) cup for the counter and 8 times \( x \) (for 8 muffins) for the muffins, and the total flour used is 4 cups.

Step 2: Form the equation

The total flour is the flour for the counter plus the flour for the muffins. So the equation is \( 8x+\frac{1}{4}=4 \), which matches equation 3.

Part (c)

Step 1: Define variables

Let \( x \) be the length of each of the four equal - length pieces. The original ribbon is 8 feet long. She cuts off \( \frac{1}{4} \) foot, and then cuts the remaining length into 4 pieces of length \( x \) each.

Step 2: Form the equation

The total length of the ribbon is the length of the cut - off piece plus the length of the four equal - length pieces. So the equation is \( \frac{1}{4}+4x = 8 \), which matches equation 1.

Answer:

s:
a) Matches with 2. \( 4+\frac{1}{4}x = 8 \)

b) Matches with 3. \( 8x+\frac{1}{4}=4 \)

c) Matches with 1. \( \frac{1}{4}+4x = 8 \)