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### Question 18:
A woman divides 60% of her income among her children $ P, Q, R $ in the ratio $ 5 : 4 : 7 $. The difference between R and Q's share is \$1800. Find the amount the woman keeps for herself.

#### Solution:
- Let the shares of $ P $, $ Q $, and $ R $ be $ 5x $, $ 4x $, and $ 7x $, respectively.
- The difference between $ R $ and $ Q $'s share is given as \$1800:
  \[
  7x - 4x = 1800 \implies 3x = 1800 \implies x = 600
  \]
- Therefore, the shares of $ P $, $ Q $, and $ R $ are:
  \[
  P = 5 \times 600 = 3000, \quad Q = 4 \times 600 = 2400, \quad R = 7 \times 600 = 4200
  \]
- Total amount distributed to the children is:
  \[
  3000 + 2400 + 4200 = 9600
  \]
- Since this is 60% of the woman's income, let her total income be $ I $. Thus, 60% of $ I $ is 9600:
  \[
  0.6I = 9600 \implies I = \frac{9600}{0.6} = 16000
  \]
- The woman keeps 40% of her income, so:
  \[
  \text{Amount she keeps} = 0.4 \times 16000 = 6400
  \]
- **Answer: C. \$6400**

---

### Question 19:
Set A has 4 numbers, and set B has 9 numbers. How many distinct $(x, y)$ points can be formed such that $x$ is selected from set A and $y$ is selected from set B, given that all these numbers are distinct?

#### Solution:
- $x$ can be selected from set A (which has 4 numbers), and $y$ can be selected from set B (which has 9 numbers).
- The number of distinct ordered pairs $(x, y)$ is simply the product of the number of choices for $x$ and $y$:
  \[
  \text{Total distinct points} = 4 \times 9 = 36
  \]
- **Answer: C. 36**

---

### Question 20:
Jane invested \$48,000 in a savings bank account that earns 10% interest compounded annually. What is the total amount at the end of 2 years?

#### Solution:
The formula for compound interest is:
\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]
Where:
- $A$ is the final amount.
- $P = 48000$ (the principal).
- $r = 0.10$ (the annual interest rate).
- $n = 1$ (since interest is compounded annually).
- $t = 2$ (the time in years).

Substituting the values:
\[
A = 48000 \left( 1 + \frac{0.10}{1} \right)^{1 \times 2} = 48000 \times (1.10)^2
\]
First calculate $ (1.10)^2 $:
\[
(1.10)^2 = 1.21
\]
Now, calculate the final amount:
\[
A = 48000 \times 1.21 = 58080
\]
- **Answer: B. \$58080**

---

Would you like a detailed explanation on any part? Here are 5 related questions you could explore:

1. What is the formula for compound interest, and how does it differ from simple interest?
2. How can ratios be used to solve problems involving distribution of income?
3. What is the significance of ordered pairs in combinatorics, and how do we calculate the number of distinct pairs?
4. What happens if interest is compounded monthly instead of annually? How would that affect the amount?
5. How can percentage-based word problems be solved using algebraic expressions?

**Tip:** When dealing with ratios in word problems, always represent unknowns using variables to set up proportional relationships easily!

Solve the following questions:

1. A woman divides her 60% of income among her children P, Q, and R into 5:4:7. If the difference between R and Q's share is $1800, find the amount that the woman keeps for herself.
2. Set A has 4 numbers and set B has 9 numbers. How many distinct (x, y) points can be formed such that x is selected from set A while y is a member of set B? Given that all these numbers are distinct.
3. Jane invested $48,000 in a savings bank account that earns 10% interest compounded annually. What's the total amount at the end of 2 years?

This page covers problems related to income distribution using ratios, combinatorics with distinct numbers, and compound interest calculations. Learn how to calculate how much a woman keeps from her income, how to find distinct pairs from two sets, and how to compute compound interest over 2 years for a bank account.

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