Question

7.3.ip-9
dennis made a table showing how much fruit he adds to each bowl of fruit salad. how many bowls does it take before dennis uses the equivalent of an entire bowl of pineapple? explain how you found the answer.
click the icon to view the table.

number of bowls	amount	less than 1 or greater than 1?
2	\\(\frac{11}{24} + \frac{11}{24} = \square\\)	\\(\boldsymbol{\bigtriangledown}\\)
3	\\(\frac{11}{24} + \frac{11}{24} + \frac{11}{24} = \square\\)	\\(\boldsymbol{\bigtriangledown}\\)
4	\\(\frac{11}{24} + \frac{11}{24} + \frac{11}{24} + \frac{11}{24} = \square\\)	\\(\boldsymbol{\bigtriangledown}\\)

click check answer.

Explanation:

Step1: Analyze the amount per bowl

Each bowl has $\frac{11}{24}$ of a bowl of pineapple. We need to find $n$ such that $n\times\frac{11}{24}\geq1$.

Step2: Solve for n

We solve the inequality $n\times\frac{11}{24}\geq1$. Multiply both sides by $\frac{24}{11}$: $n\geq\frac{24}{11}\approx2.18$. Since $n$ must be an integer, we test $n = 3$? Wait, no, wait. Wait, first let's check the cumulative amounts:

For $n = 1$: $\frac{11}{24}\approx0.458 < 1$

For $n = 2$: $\frac{11}{24}+\frac{11}{24}=\frac{22}{24}=\frac{11}{12}\approx0.916 < 1$

For $n = 3$: $\frac{11}{24}\times3=\frac{33}{24}=\frac{11}{8}=1.375 > 1$

Wait, but the question is "how many bowls does it take before Dennis uses the equivalent of an entire bowl of pineapple". So we need to find the smallest $n$ where the total is at least 1.

Wait, when $n = 2$, total is $\frac{22}{24}=\frac{11}{12}<1$. When $n = 3$, total is $\frac{33}{24}=\frac{11}{8}=1.375\geq1$. So we need to check the cumulative sums:

First, let's compute the amounts:

For 1 bowl: $\frac{11}{24}$ (less than 1)

For 2 bowls: $\frac{11}{24}+\frac{11}{24}=\frac{22}{24}=\frac{11}{12}$ (less than 1, since $\frac{11}{12}\approx0.916 < 1$)

For 3 bowls: $\frac{11}{24}\times3=\frac{33}{24}=\frac{11}{8}=1.375$ (greater than 1)

So before (i.e., the number of bowls when the total becomes at least 1) – wait, the wording is "how many bowls does it take before Dennis uses the equivalent of an entire bowl of pineapple". So "before" might mean when the total reaches or exceeds 1. Wait, maybe the question is how many bowls until the total is at least 1. So we need to find the smallest $n$ where $\sum_{k = 1}^n\frac{11}{24}\geq1$.

We can solve $n\times\frac{11}{24}\geq1\implies n\geq\frac{24}{11}\approx2.18$. So the smallest integer $n$ is 3? Wait, no, wait: when $n = 2$, total is $\frac{22}{24}<1$. When $n = 3$, total is $\frac{33}{24}>1$. So the number of bowls it takes before (i.e., when the total is at least 1) is 3? Wait, but let's re - read the question: "how many bowls does it take before Dennis uses the equivalent of an entire bowl of pineapple". So "before" might be a bit ambiguous, but in terms of cumulative: after 2 bowls, he has used $\frac{22}{24}$, which is less than 1. After 3 bowls, he has used $\frac{33}{24}$, which is more than 1. So the number of bowls it takes to use at least 1 bowl of pineapple is 3? Wait, but let's check the table:

Wait, the table has columns: Number of Bowls, Amount, Less than 1 or Greater than 1?

For 1 bowl: Amount $\frac{11}{24}$, Less than 1.

For 2 bowls: $\frac{11}{24}+\frac{11}{24}=\frac{22}{24}=\frac{11}{12}$, Less than 1.

For 3 bowls: $\frac{11}{24}\times3=\frac{33}{24}=\frac{11}{8}=1.375$, Greater than 1.

So the question is "how many bowls does it take before Dennis uses the equivalent of an entire bowl of pineapple". So we need to find when the total is at least 1. So after 2 bowls, it's less than 1. After 3 bowls, it's more than 1. So the number of bowls is 3? Wait, but let's confirm the calculation:

We need to find $n$ such that $n\times\frac{11}{24}\geq1$. Solving for $n$: $n\geq\frac{24}{11}\approx2.18$. Since $n$ must be an integer, the smallest $n$ is 3.

Answer:

3