Marie can eat the entire cake in 10 minutes. Antoinette joins her after 3 minutes, and together they eat the rest of the cake in 4 minutes. How long would it have taken Antoinette to eat the entire cake alone

To solve this problem, we need to determine how long it would have taken Antoinette to eat the entire cake alone, based on the information provided.

### Step 1: Define the rates of Marie and Antoinette

- Marie can eat the entire cake in 10 minutes. Therefore, Marie’s rate of eating the cake is:
  \[
  \text{Rate of Marie} = \frac{1 \text{ cake}}{10 \text{ minutes}} = \frac{1}{10} \text{ cakes per minute}.
  \]

- Let Antoinette’s unknown time to eat the entire cake be \( T \) minutes. Therefore, Antoinette’s rate of eating the cake is:
  \[
  \text{Rate of Antoinette} = \frac{1}{T} \text{ cakes per minute}.
  \]

### Step 2: Account for the time when Marie is eating alone
Marie eats alone for the first 3 minutes. In that time, she eats:
\[
\text{Cake eaten by Marie alone} = 3 \times \frac{1}{10} = \frac{3}{10} \text{ of the cake}.
\]
This means \(\frac{7}{10}\) of the cake is left to be eaten when Antoinette joins.

### Step 3: Both eat together for 4 minutes
When Antoinette joins, the two together eat the remaining \(\frac{7}{10}\) of the cake in 4 minutes. So the combined rate of Marie and Antoinette is:
\[
\text{Combined rate} = \frac{7}{10} \text{ cakes in 4 minutes} = \frac{7}{10} \div 4 = \frac{7}{40} \text{ cakes per minute}.
\]

### Step 4: Express the combined rate in terms of both their rates
We know the combined rate is the sum of Marie’s rate and Antoinette’s rate:
\[
\frac{1}{10} + \frac{1}{T} = \frac{7}{40}.
\]
Now, solve for \( T \):

\[
\frac{1}{10} + \frac{1}{T} = \frac{7}{40}
\]
Subtract \(\frac{1}{10}\) from both sides:
\[
\frac{1}{T} = \frac{7}{40} - \frac{1}{10}.
\]
Convert \(\frac{1}{10}\) to have a denominator of 40:
\[
\frac{1}{10} = \frac{4}{40}.
\]
So:
\[
\frac{1}{T} = \frac{7}{40} - \frac{4}{40} = \frac{3}{40}.
\]
Therefore:
\[
T = \frac{40}{3} \approx 13.33 \text{ minutes}.
\]

### Final Answer:
It would have taken Antoinette approximately 13.33 minutes to eat the entire cake alone.

Would you like further details or have any questions?

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Here are 5 related questions to further your understanding:

1. If both started eating the cake together, how long would it take them to finish?
2. If Antoinette ate the first 3 minutes alone, how long would it take Marie to finish the rest of the cake?
3. What if Marie could eat the cake in 8 minutes instead of 10? How would that change the time for Antoinette?
4. If a third person joined after 5 minutes, how would their combined rate affect the time to finish the cake?
5. If they ate the cake at varying rates over time, how would you set up an equation to model that?

**Tip:** When working with rates, always remember that the sum of individual rates gives the combined rate when people work together!

Marie can eat the entire cake in 10 minutes. Antoinette joins her after 3 minutes, and together they eat the rest of the cake in 4 minutes. How long would it have taken Antoinette to eat the entire cake alone?

Find out how long it would take Antoinette to eat the entire cake alone if Marie can finish it in 10 minutes, but Antoinette joins her after 3 minutes. This problem involves rates, algebraic equations, and work-time concepts, suitable for Grades 8-10.

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