Let's go through the question step by step to find the correct equation and statements.
Leena consumes:
- 400 calories at breakfast
- 350 calories at lunch
Her total calories for the day is reported to be 1,500 calories. Let [tex]\( x \)[/tex] represent the calories consumed at dinner. According to the problem, Leena consumes [tex]\(\frac{2}{3}\)[/tex] of [tex]\( x \)[/tex] calories.
First, let's calculate the combined calories for breakfast and lunch:
[tex]\[
\text{Calories from breakfast and lunch} = 400 + 350 = 750 \text{ calories}
\][/tex]
The total calories consumed in a day:
[tex]\[
1,500 \text{ calories} = 750 \text{ calories (breakfast and lunch)} + \frac{2}{3} x \text{ (dinner calories)}
\][/tex]
We need to solve the equation for [tex]\( x \)[/tex]:
[tex]\[
1,500 = 750 + \frac{2}{3} x
\][/tex]
Isolating [tex]\( \frac{2}{3} x \)[/tex]:
[tex]\[
1,500 - 750 = \frac{2}{3} x
\][/tex]
Simplifying the left side:
[tex]\[
750 = \frac{2}{3} x
\][/tex]
To solve for [tex]\( x \)[/tex], we multiply both sides by the reciprocal of [tex]\(\frac{2}{3}\)[/tex], which is [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
x = 750 \times \frac{3}{2} = 1125 \text{ calories}
\][/tex]
Thus, Leena consumed 1,125 calories at dinner.
Now we can assess the given statements:
1. "Leena consumed 1,500 calories at dinner." This statement is incorrect because she consumed 1,500 calories in total, not just at dinner.
2. "The equation [tex]\(\frac{2}{3}(x + 400 + 350) = x\)[/tex] can be used to model the situation." This equation does not correctly represent the given situation. Therefore, this statement is incorrect.
3. "Leena consumed 500 calories at dinner." This statement is incorrect because we found that she consumed 1,125 calories at dinner.
4. "The equation [tex]\(\frac{2}{3}(x) = x (400 + 300)\)[/tex] can be used to model the situation." This is not mathematically correct for the given situation, and it mixes the variables and constants inappropriately. Therefore, this statement is incorrect.
5. "Leena consumed 1,000 calories at dinner." This statement is incorrect because she consumed 1,125 calories at dinner.
6. "The equation [tex]\(\frac{2}{3} x (400 + 300) = x\)[/tex] can be used to model the situation." This is also incorrect because it does not accurately represent the given problem's conditions.
The correct answer based on the detailed step-by-step solution is:
- None of the provided statements accurately describe the equations or the situation correctly.
