found $$ \begin{array}{l} ext { 14. Higher Order Thinking Leah wrote } \ 2 ext { different fractions with the same } \ ext { denominator. Both fractions were les } \ ext { than 1. Can their sum equal 1? Can the } \ ext { sum be greater than 1? Explain. }\end{array} $$

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found $$ \begin{array}{l}	ext { 14. Higher Order Thinking Leah wrote } \ 2 	ext { different fractions with the same } \ 	ext { denominator. Both fractions were les } \ 	ext { than 1. Can their sum equal 1? Can the } \ 	ext { sum be greater than 1? Explain. }\end{array} $$

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To solve Leah's problem, we need to analyze the conditions given: 1. Leah wrote two different fractions with the same denominator. 2. Both fractions are less than 1. 3. We need to determine if their sum can equal 1 or be greater than 1. Let's denote the two fractions as: $$ \frac{a}{d} \quad \text{and} \quad \frac{b}{d} $$ where $$ a $$ and $$ b $$ are the numerators, and $$ d $$ is the common denominator. Since both fractions are less than 1, we have: $$ a < d \quad \text{and} \quad b < d $$ ### Step 1: Can their sum equal 1? The sum of the two fractions is: $$ \frac{a}{d} + \frac{b}{d} = \frac{a + b}{d} $$ For this sum to equal 1, we need: $$ \frac{a + b}{d} = 1 $$ This implies: $$ a + b = d $$ Since both $$ a $$ and $$ b $$ are less than $$ d $$, the maximum value of $$ a + b $$ would be less than $$ d + d = 2d $$. However, since both fractions are less than 1, the maximum value of $$ a + b $$ must also be less than $$ d $$. Therefore, it is possible for $$ a + b $$ to equal $$ d $$ if $$ a $$ and $$ b $$ are chosen appropriately, such as $$ a = d - b $$ where $$ b < d $$. ### Conclusion for sum equal to 1: Yes, the sum of the two fractions can equal 1 if $$ a + b = d $$. ### Step 2: Can the sum be greater than 1? For the sum to be greater than 1, we need: $$ \frac{a + b}{d} > 1 $$ This implies: $$ a + b > d $$ However, since both $$ a $$ and $$ b $$ are less than $$ d $$, the maximum value of $$ a + b $$ is less than $$ d + d = 2d $$. But since both fractions are less than 1, $$ a + b $$ cannot exceed $$ d $$. Therefore, it is impossible for the sum of the two fractions to be greater than 1. ### Conclusion for sum greater than 1: No, the sum of the two fractions cannot be greater than 1. ### Final Summary: - The sum of the two fractions can equal 1 if $$ a + b = d $$. - The sum of the two fractions cannot be greater than 1. Yes, the sum of the two fractions can equal 1 if their numerators add up to the common denominator. However, the sum cannot be greater than 1 because both fractions are less than 1.

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