(8) Assessment Practice - Kate's dad gives each of his 5 children an equal part of $$ \$ 340 $$ to buy gifts. Kate adds 528 to her portion. She finds classic DVDs for $$ \$ 8 $$ each. How many DVDs can Kate buy? Explain how you solve, Use one or more equations and bar diagrams in your explanation. Tell what your variables represent. (

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To solve the problem, we will follow these steps:

1. **Extract known conditions:**
   - Total amount of money: $$ \$ 340 $$
   - Number of children: $$ 5 $$
   - Additional amount Kate adds to her portion: $$ \$ 528 $$
   - Cost of each DVD: $$ \$ 8 $$

2. **Define variables:**
   - Let $$ x $$ be the amount of money each child receives from their dad.
   - Let $$ y $$ be the total amount of money Kate has after adding her portion and the additional amount.

3. **Calculate the amount each child receives:**
   Since the total amount is divided equally among the 5 children, we can express $$ x $$ as:
   $$
   x = \frac{340}{5}
   $$

4. **Calculate Kate's total amount:**
   Kate's total amount $$ y $$ can be expressed as:
   $$
   y = x + 528
   $$

5. **Determine how many DVDs Kate can buy:**
   The number of DVDs $$ n $$ that Kate can buy is given by:
   $$
   n = \frac{y}{8}
   $$

Now, let's perform the calculations step by step.

### Step 1: Calculate $$ x $$
$$
x = \frac{340}{5}
$$

### Step 2: Calculate $$ y $$
$$
y = x + 528
$$

### Step 3: Calculate $$ n $$
$$
n = \frac{y}{8}
$$

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{340}{5}$$
- step1: Reduce the numbers:
  $$\frac{68}{1}$$
- step2: Calculate:
  $$68$$
Calculate or simplify the expression $$ (340/5) + 528 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{340}{5}\right)+528$$
- step1: Divide the terms:
  $$68+528$$
- step2: Add the numbers:
  $$596$$
Calculate or simplify the expression $$ ((340/5) + 528)/8 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(\left(\frac{340}{5}\right)+528\right)}{8}$$
- step1: Remove the parentheses:
  $$\frac{\left(\frac{340}{5}\right)+528}{8}$$
- step2: Divide the terms:
  $$\frac{68+528}{8}$$
- step3: Add the numbers:
  $$\frac{596}{8}$$
- step4: Reduce the fraction:
  $$\frac{149}{2}$$
Now that we have performed the calculations, let's summarize the results:

1. **Amount each child receives ($$ x $$):**
   $$
   x = 68
   $$
   Each child receives $$ \$ 68 $$.

2. **Total amount Kate has after adding her portion ($$ y $$):**
   $$
   y = 68 + 528 = 596
   $$
   Kate has $$ \$ 596 $$.

3. **Number of DVDs Kate can buy ($$ n $$):**
   $$
   n = \frac{596}{8} = 74.5
   $$
   Since Kate cannot buy half a DVD, she can buy $$ 74 $$ DVDs.

### Conclusion:
- **Variables:**
  - $$ x $$: Amount each child receives from their dad (\$68).
  - $$ y $$: Total amount Kate has after adding her portion (\$596).
  - $$ n $$: Number of DVDs Kate can buy (74 DVDs).

### Explanation:
- Kate's dad gives each child an equal share of the total amount, which is calculated by dividing $$ \$ 340 $$ by $$ 5 $$.
- After receiving her share, Kate adds $$ \$ 528 $$ to it, giving her a total of $$ \$ 596 $$.
- Finally, to find out how many DVDs she can buy, we divide her total amount by the cost of each DVD, which is $$ \$ 8 $$. Since she can only buy whole DVDs, she can purchase $$ 74 $$ DVDs.
Kate can buy 74 DVDs.

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