Evan ate three-eighths of a hoagie for dinner. On the next day, he ate two-thirds of what was left for lunch. How much of the entire hoagie did Evan eat for lunch? Which process will you use to solve this problem?

a. 3/8÷2/3=9/16
b. 3/8+2/3=2/5/24=1 1/24
c. 3/8x2/3=6÷6/24÷6=1/4
d. 5/8x2/3=10÷2/24÷2=5/12
e. 5/8÷2/3=15/16
pick multiple answers

11 answers

To solve the problem, we need to follow these steps:

  1. Determine how much of the hoagie is left after Evan ate three-eighths of it.
  2. Calculate how much of what was left he ate for lunch.

Step 1: Calculate what is left after dinner.

Evan ate \( \frac{3}{8} \) of the hoagie. The amount left is: \[ 1 - \frac{3}{8} = \frac{8}{8} - \frac{3}{8} = \frac{5}{8} \]

Step 2: Calculate how much he ate for lunch.

Evan ate two-thirds of what was left, which is: \[ \frac{2}{3} \times \frac{5}{8} \]

Now we perform the multiplication: \[ \frac{2 \times 5}{3 \times 8} = \frac{10}{24} \]

This fraction can be simplified: \[ \frac{10}{24} = \frac{5}{12} \]

Thus, Evan ate \( \frac{5}{12} \) of the entire hoagie for lunch.

Options Analysis:

Looking at the choices provided:

  • (a) \( \frac{3}{8} ÷ \frac{2}{3} \) does not apply, as it does not calculate the amount eaten for lunch.
  • (b) \( \frac{3}{8} + \frac{2}{3} \) is incorrect, since we do not add but rather multiply the remaining amount.
  • (c) \( \frac{3}{8} \times \frac{2}{3} \) is incorrect because it does not represent the calculation for lunch. Instead—it represents something different and is used incorrectly.
  • (d) \( \frac{5}{8} \times \frac{2}{3} = \frac{10}{24} = \frac{5}{12} \) is the correct choice, as it correctly finds how much Evan ate for lunch.
  • (e) \( \frac{5}{8} ÷ \frac{2}{3} \) is also incorrect, as it would not calculate the amount eaten, but a different operation.

Final Answers:

The correct process is (d) \( \frac{5}{8} \times \frac{2}{3} = \frac{5}{12} \).

Only choice (d) is appropriate in solving how much of the entire hoagie Evan ate for lunch.

there is only one answer?

Apologies for the oversight in my earlier response! You are correct to seek clarity regarding whether there could be multiple applicable answers based on the choices provided.

Based on how we can interpret the choices in terms of the steps required to solve the problem:

  • (a) \( \frac{3}{8} ÷ \frac{2}{3} = \frac{9}{16} \): This operation does not give the right answer for lunch and does not reflect the problem correctly.
  • (b) \( \frac{3}{8} + \frac{2}{3} \): This addition does not apply to the scenario since we are finding a portion of what is left, not combining two fractions.
  • (c) \( \frac{3}{8} \times \frac{2}{3} = \frac{6}{24} = \frac{1}{4} \): This is also incorrect in representing the problem as we need to use the remaining amount \( \frac{5}{8} \) instead of the eaten portion.
  • (d) \( \frac{5}{8} \times \frac{2}{3} = \frac{10}{24} = \frac{5}{12} \): This is the correct method for solving how much of the entire hoagie Evan ate for lunch.
  • (e) \( \frac{5}{8} ÷ \frac{2}{3} \): This also does not provide the right solution and is irrelevant to the problem asked.

Thus, the only correct answer for how much Evan ate for lunch is (d) \( \frac{5}{12} \). Thank you for your patience, and I hope this clarifies the solution!

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Eduardo used a number line model shown to model an equation.
which of the following could be Eduardo's equation
a.-5-3=-8
b.-5+8=3
c. 5+(-8)=-3
d. 5+(-3)=2

To analyze which of the provided equations could correspond to the number line model shown, it helps to understand what each equation is doing in terms of movement along the number line.

Given the number line and its values, we can interpret the following:

  1. Eduardo appears to start at -5.
  2. It seems to move to the right towards positive numbers (indicated by the arrow going to the right).

Let's examine each equation:

a. \(-5 - 3 = -8\)

  • This indicates starting from -5 and moving left (subtracting 3) to -8. This would be a movement left, which contradicts the rightward movement indicated in the diagram. Not a match.

b. \(-5 + 8 = 3\)

  • This indicates starting from -5 and moving right by 8 units, landing at 3. This represents a considerable rightward movement, making this a good candidate. Possible match.

c. \(5 + (-8) = -3\)

  • This indicates starting from 5 and moving left by 8 units to reach -3. This does not fit the starting value of -5, nor does it suggest movement right from -5. Not a match.

d. \(5 + (-3) = 2\)

  • This indicates starting from 5 and moving left by 3 to reach 2. This also does not correspond to starting at -5 and moving right. Not a match.

Conclusion:

The only equation that matches Eduardo's possible movement along the number line, starting from -5 and moving to the right, is (b) \(-5 + 8 = 3\).

select all expressions that are equivalent to this algebraic expression
(2x-1)-3.25(x+3)
a. 2x-1-3.25x-9.75
b.2x-3.25x-1-9.75
c. -1.25x-1-9.75
d.-1.25x-10.75
e.2x-1-3.25x+9.75
f.1.25x+10.75
g.-1.25x-4

Let's first simplify the given algebraic expression:

\[ (2x - 1) - 3.25(x + 3) \]

  1. Start by distributing \(-3.25\) throughout the expression \((x + 3)\):

\[ (2x - 1) - (3.25x + 9.75) \]

This simplifies to:

\[ 2x - 1 - 3.25x - 9.75 \]

  1. Now, combine like terms:
  • Combine the \(x\) terms: \(2x - 3.25x = -1.25x\)
  • Combine the constant terms: \(-1 - 9.75 = -10.75\)

The simplified expression is:

\[ -1.25x - 10.75 \]

Now, let's evaluate each option to determine which expressions are equivalent to \(-1.25x - 10.75\).

Check each option:

  • (a) \(2x - 1 - 3.25x - 9.75 = -1.25x - 10.75\) (This is equivalent)
  • (b) \(2x - 3.25x - 1 - 9.75 = -1.25x - 10.75\) (This is equivalent)
  • (c) \(-1.25x - 1 - 9.75 = -1.25x - 10.75\) (This is equivalent)
  • (d) \(-1.25x - 10.75 = -1.25x - 10.75\) (This is equivalent)
  • (e) \(2x - 1 - 3.25x + 9.75 = -1.25x + 8.75\) (Not equivalent)
  • (f) \(1.25x + 10.75\) (Not equivalent)
  • (g) \(-1.25x - 4\) (Not equivalent)

Conclusion:

The expressions that are equivalent to the original expression \((2x - 1) - 3.25(x + 3)\) are:

  • (a)
  • (b)
  • (c)
  • (d)

Thus, options a, b, c, and d are all equivalent to the simplified expression.

select all expressions that are equivalent to this algebraic expression
(3x-8)+(2x+5)-(4x-8)
a. 3x-8+2x+5-4x-8
b. 3x-8+2x+5-4x+8
c. 3x+2x-4x-8+5+8
d. 5x-4x-8+5+8
e.x-8+5+8
f.x-13+8
g.x-3+8
h.x-3+8
i.x-21
j.x+5

Let's simplify the given algebraic expression step by step:

\[ (3x - 8) + (2x + 5) - (4x - 8) \]

  1. Distribute the negative sign into \(-(4x - 8)\):

\[ (3x - 8) + (2x + 5) - 4x + 8 \]

  1. Now, combine like terms:
    • Combine the \(x\) terms:
      • \(3x + 2x - 4x = (3 + 2 - 4)x = x\)
    • Combine the constant terms:
      • \(-8 + 5 + 8 = (-8 + 8 + 5) = 5\)

So when simplified, the algebraic expression becomes:

\[ x + 5 \]

Now, we need to evaluate each option to determine which expressions are equivalent to \(x + 5\).

Check each option:

  • (a) \(3x - 8 + 2x + 5 - 4x - 8\)

    • Combine like terms:
    • \( (3x + 2x - 4x) + (-8 + 5 - 8) = x - 11\) (Not equivalent)
  • (b) \(3x - 8 + 2x + 5 - 4x + 8\)

    • Combine like terms:
    • \( (3x + 2x - 4x) + (-8 + 5 + 8) = x + 5\) (Equivalent)
  • (c) \(3x + 2x - 4x - 8 + 5 + 8\)

    • Combine like terms:
    • \( (3x + 2x - 4x) + (-8 + 5 + 8) = x + 5\) (Equivalent)
  • (d) \(5x - 4x - 8 + 5 + 8\)

    • Combine like terms:
    • \( (5x - 4x) + (-8 + 5 + 8) = x + 5\) (Equivalent)
  • (e) \(x - 8 + 5 + 8\)

    • Combine like terms:
    • \( x + (-8 + 5 + 8) = x + 5\) (Equivalent)
  • (f) \(x - 13 + 8\)

    • Combine like terms:
    • \( x + (-13 + 8) = x - 5\) (Not equivalent)
  • (g) \(x - 3 + 8\)

    • Combine like terms:
    • \( x + (-3 + 8) = x + 5\) (Equivalent)
  • (h) \(x - 3 + 8\) (Same as g, thus equivalent)

    • Combine like terms:
    • \(x + (-3 + 8) = x + 5\) (Equivalent)
  • (i) \(x - 21\) (Not equivalent)

  • (j) \(x + 5\) (Equivalent)

Conclusion:

The expressions that are equivalent to \(x + 5\) are:

  • (b)
  • (c)
  • (d)
  • (e)
  • (g)
  • (h)
  • (j)

Thus, the correct options are (b), (c), (d), (e), (g), (h), and (j).

select all expressions that are equivalent to this algebraic expression
5(3x+9)
a. 8x+14
b. 5x3x+5x9
c. 5x3x+9
d. 15x+9
e. 15x+45

Let's simplify the given algebraic expression step by step:

Step 1: Distribute

The expression \(5(3x + 9)\) can be simplified by distributing the 5:

\[ 5(3x) + 5(9) = 15x + 45 \]

So the simplified expression is:

\[ 15x + 45 \]

Now we will review each option to determine which expressions are equivalent to \(15x + 45\).

Check each option:

  • (a) \(8x + 14\)

    • Not equivalent. (This does not match the form of the simplified expression.)
  • (b) \(5 \times 3x + 5 \times 9\)

    • This is equivalent to \(15x + 45\) after simplification. (This is equivalent. You could also write this as \(15x + 45\).)
  • (c) \(5x \dots 3x + 9\)

    • This expression is unclear due to notation. Assuming it means \(5x \cdot 3x + 9\) (which is \(15x^2 + 9\), it is not equivalent).
  • (d) \(15x + 9\)

    • Not equivalent. (It does not match \(15x + 45\).)
  • (e) \(15x + 45\)

    • Equivalent. (This is exactly what we simplified to.)

Conclusion:

The expressions that are equivalent to \(5(3x + 9)\) are:

  • (b) \(5 \times 3x + 5 \times 9\)
  • (e) \(15x + 45\)

Thus, the correct options are (b) and (e).