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Let's define variables to represent the number of pennies each person has before losing them:

- Let \( d \) be the number of pennies Dante originally had.
- Let \( m \) be the number of pennies Mia originally had.

We know from the problem that together, Dante and Mia have a total of 350 pennies, which gives us the equation:

\[
d + m = 350
\]

Next, the problem states that Dante lost \( \frac{1}{2} \) of his pennies, so after losing his pennies, Dante has \( \frac{d}{2} \) pennies left. Similarly, Mia lost \( \frac{1}{3} \) of her pennies, so after losing her pennies, Mia has \( \frac{2}{3} m \) pennies left.

After losing their pennies, they both have an equal number of pennies:

\[
\frac{d}{2} = \frac{2}{3} m
\]

We now have a system of two equations:

1. \( d + m = 350 \)
2. \( \frac{d}{2} = \frac{2}{3} m \)

### Step 1: Solve the second equation for \( d \)
Multiply both sides of the second equation by 2 to eliminate the fraction:

\[
d = \frac{4}{3} m
\]

### Step 2: Substitute this expression for \( d \) into the first equation
Substitute \( d = \frac{4}{3} m \) into \( d + m = 350 \):

\[
\frac{4}{3} m + m = 350
\]

Combine terms:

\[
\frac{4}{3} m + \frac{3}{3} m = 350
\]

\[
\frac{7}{3} m = 350
\]

### Step 3: Solve for \( m \)
Multiply both sides by 3 to get rid of the denominator:

\[
7m = 1050
\]

Now, divide by 7:

\[
m = 150
\]

### Step 4: Find \( d \)
Now that we know \( m = 150 \), substitute this value into the equation \( d + m = 350 \):

\[
d + 150 = 350
\]

\[
d = 200
\]

### Step 5: Calculate the number of pennies lost
Dante lost \( \frac{1}{2} \) of his pennies, so he lost:

\[
\frac{1}{2} \times 200 = 100
\]

Mia lost \( \frac{1}{3} \) of her pennies, so she lost:

\[
\frac{1}{3} \times 150 = 50
\]

### Step 6: Total pennies lost
Altogether, Dante and Mia lost:

\[
100 + 50 = 150 \text{ pennies}
\]

Thus, they lost **150 pennies** in total.

---

Would you like more details on any step, or do you have any questions? Here are five related questions to explore:

1. What if Dante had lost \( \frac{1}{3} \) of his pennies and Mia \( \frac{1}{2} \) of hers?
2. How would the solution change if they had 500 pennies in total?
3. Can you verify the solution using substitution back into the original equations?
4. What would happen if both lost the same fraction of their pennies?
5. How would the problem change if Dante and Mia started with different total numbers of pennies?

**Tip:** When solving systems of equations with fractions, it's helpful to eliminate the fractions early to simplify calculations.

Together, Dante and Mia have a total of 350 pennies in their piggy banks. After Dante lost 1/2 of his pennies and Mia lost 1/3 of her pennies, they both had an equal number of pennies. Altogether, how many pennies did they lose?

Learn how to solve a system of equations where Dante and Mia lose a fraction of their pennies. Together, they start with 350 pennies, and after losing 1/2 and 1/3 of their pennies, they end up with an equal amount. This algebraic problem is suitable for students in grades 6-8.

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