Write in figures: One and a half million. 2) Express % as a decimal. 3) Expand 72 x 53. 4) Which of these years was a leap year: 2020, 2021, 2022?

One million is 1,000,000.

Half a million is $\frac{1}{2} \times 1,000,000 = 500,000$.

$1,000,000 + 500,000 = 1,500,000$.
In figures, one and a half million is $\boxed{1,500,000}$.

2) Fractions can be expressed as decimals. Express % as a decimal
This question asks how to express a percentage as a decimal in general.

For example, if you have $X\%$, it is equivalent to $\frac{X}{100}$ as a decimal.
So, to express a percentage as a decimal, you divide the number by 100. For example, $25\% = \frac{25}{100} = \boxed{0.25}$.

3) Numbers written in index form can be expanded. Expand $7^2 \times 5^3$

$7^2 = 7 \times 7 = 49$.

$5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125$.

$49 \times 125 = 6125$.
The expanded form is $\boxed{49 \times 125 = 6125}$.

4) Some years are called leap years. Which of these years was a leap year: 2020, 2021, 2022?

A year is a leap year if it is divisible by 4, unless it is a century year (like 1900) which must be divisible by 400.

• For 2020: $2020 \div 4 = 505$. Since 2020 is divisible by 4, it is a leap year.
• For 2021: $2021 \div 4 = 505$ with a remainder of 1. So, 2021 is not a leap year.
• For 2022: $2022 \div 4 = 505$ with a remainder of 2. So, 2022 is not a leap year. The leap year among the given years is $\boxed{2020}$.

5) A mixed fraction can be converted to an improper fraction. Convert $14 \frac{1}{2}$ to an improper fraction.

$14 \times 2 = 28$.

$28 + 1 = 29$.

The improper fraction is $\boxed{\frac{29}{2}}$.

6) Write 16.53675 correct to 4 significant figures.

These are 1, 6, 5, 3.

The number becomes 16.54.
16.53675 correct to 4 significant figures is $\boxed{16.54}$.

7) Given that $x = 5$ and $y = 4$, calculate $(x+y)^2 + (2x-y)^2$

The value of the expression is $\boxed{117}$.

8) The sum of two numbers is 502. If one of the numbers is 128, what is the value of the other number?

Given $a + b = 502$.

Substitute this into the equation.
$128 + b = 502$.

$b = 502 - 128$.
$b = 374$.
The value of the other number is $\boxed{374}$.

9) Modular arithmetic deals with the remainder. What is $3 + 7 \pmod{4}$?

$3 + 7 = 10$.

$10 \div 4 = 2$ with a remainder of $10 - (4 \times 2) = 10 - 8 = 2$.
So, $10 \equiv 2 \pmod{4}$.
$3 + 7 \pmod{4}$ is $\boxed{2}$.

10) A clock loses 2 minutes in every 3 hours. How many minutes does it lose in 15 hours?

Loss rate = $\frac{2 \text{ minutes}}{3 \text{ hours}}$.

Total loss = Loss rate $\times$ Total hours

The clock loses $\boxed{10 \text{ minutes}}$ in 15 hours.

11) After spending $\frac{1}{4}$ of my money on bread and $\frac{1}{3}$ on eggs, what fraction have I left?

Total spent = $\frac{1}{4} + \frac{1}{3}$.

Subtract the spent fraction from 1 to find the fraction left.

The fraction of money left is $\boxed{\frac{5}{12}}$.

12) If $\xi = \{1, 2, 3, 4, 5, 6, ..., 10\}$, $A = \{1, 3, 6, 9, 10\}$ and $B = \{2, 4, 6, 8, 10\}$, Find $(A \cap B)'$

$\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.

This includes elements common to both A and B.
$A = \{1, 3, 6, 9, 10\}$
$B = \{2, 4, 6, 8, 10\}$
The common elements are 6 and 10.
$A \cap B = \{6, 10\}$.

This includes all elements in the universal set $\xi$ that are not in $(A \cap B)$.
Elements in $\xi$: $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
Elements in $A \cap B$: $\{6, 10\}$
Remove $\{6, 10\}$ from $\xi$:
$(A \cap B)' = \{1, 2, 3, 4, 5, 7, 8, 9\}$.
$(A \cap B)' = \boxed{\{1, 2, 3, 4, 5, 7, 8, 9\}}$.