Which of these years was a leap year: 2020, 2021, 2022?

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for questions one to twelve:

1) Numbers can either be written in figures, words or Roman numerals. Write in figures: One and a half million

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

2) Fractions can be expressed as decimals. Express % as a decimal
To express a percentage as a decimal, divide the percentage value by 100.
For example, $X\%$ 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 $7^2 \times 5^3$
Expand $7^2$ .
$7^2 = 7 \times 7 = 49$
Expand $5^3$ .
$5^3 = 5 \times 5 \times 5 = 125$
Multiply the expanded values.
$49 \times 125 = 6125$
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 leap year is generally divisible by 4.
Check each year
- $2020 \div 4 = 505$ . 2020 is divisible by 4, so it is a leap year.
- $2021 \div 4 = 505$ with a remainder of 1. Not a leap year.
- $2022 \div 4 = 505$ with a remainder of 2. 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.
Multiply the whole number by the denominator.
$14 \times 2 = 28$
Add the numerator to the result.
$28 + 1 = 29$
Place this sum over the original denominator.
The improper fraction is $\boxed{\frac{29}{2}}$ .

6) Write 16.53675 correct to 4 significant figures.
Identify the first four significant figures
1, 6, 5, 3.
Look at the fifth significant figure, which is 6.
Since 6 is 5 or greater, round up the fourth significant figure (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 $(x+y)^2 \div (2x-y)^2$
Substitute $x=5$ and $y=4$ into the first part of the expression.
$(x+y)^2 = (5+4)^2 = 9^2 = 81$
Substitute $x=5$ and $y=4$ into the second part of the expression.
$(2x-y)^2 = (2 \times 5 - 4)^2 = (10 - 4)^2 = 6^2 = 36$
Perform the division.
$(x+y)^2 \div (2x-y)^2 = 81 \div 36$ $\frac{81}{36} = \frac{9 \times 9}{4 \times 9} = \frac{9}{4}$
As a decimal, $\frac{9}{4} = 2.25$ .
The value of the expression is $\boxed{\frac{9}{4} \text{ or } 2.25}$ .

8) The sum of two numbers is 502. If one of the numbers is 128, what is the value of the other number?
Let the other number be $N$ .
Set up the equation.
$128 + N = 502$
Solve for $N$ .
$N = 502 - 128$ $N = 374$
The value of the other number is $\boxed{374}$ .

9) Modular arithmetic deals with the remainder. What is $3 + 7 \pmod{4}$ ?
Perform the addition.
$3 + 7 = 10$
Find the remainder when 10 is divided by 4.
$10 \div 4 = 2 \text{ with a remainder of } 2$
So, $3 + 7 \pmod{4}$ is $\boxed{2}$ .

10) A clock loses 2 minutes in every 3 hours. How many minutes does it lose in 15 hours?
Determine the rate of loss per hour.
$\text{Rate of loss} = \frac{2 \text{ minutes}}{3 \text{ hours}}$
Multiply the rate by the total hours.
$\text{Total loss} = \frac{2}{3} \text{ minutes/hour} \times 15 \text{ hours}$ $\text{Total loss} = \frac{2 \times 15}{3} \text{ minutes}$ $\text{Total loss} = \frac{30}{3} \text{ minutes}$ $\text{Total loss} = 10 \text{ minutes}$
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?
Find the total fraction of money spent.
$\text{Total spent} = \frac{1}{4} + \frac{1}{3}$
Find a common denominator, which is 12.
$\frac{1}{4} = \frac{3}{12}$ $\frac{1}{3} = \frac{4}{12}$
Add the fractions.
$\text{Total spent} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$
Subtract the spent fraction from the total (1, or $\frac{12}{12}$ ).
$\text{Fraction left} = 1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}$
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 $(A \cap B)'$
List the universal set $\xi$ .
$\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
Find the intersection of sets A and B ( $A \cap B$ ), which are the elements common to both sets.
$A = \{1, 3, 6, 9, 10\}$ $B = \{2, 4, 6, 8, 10\}$ $A \cap B = \{6, 10\}$
Find the complement of $(A \cap B)$ , denoted as $(A \cap B)'$ .
This includes all elements in $\xi$ that are not in $(A \cap B)$ .
$(A \cap B)' = \{1, 2, 3, 4, 5, 7, 8, 9\}$
$(A \cap B)' = \boxed{\{1, 2, 3, 4, 5, 7, 8, 9\}}$ .

Which of these years was a leap year: 2020, 2021, 2022?

Step-by-step solution

One million is 1,000,000.

Half a million is 500,000.

Add these values.

Expand $7^2$ .

Expand $5^3$ .

Multiply the expanded values.

A leap year is generally divisible by 4.

Check each year

Multiply the whole number by the denominator.

Add the numerator to the result.

Place this sum over the original denominator.

Identify the first four significant figures

Look at the fifth significant figure, which is 6.

Since 6 is 5 or greater, round up the fourth significant figure (3).

Substitute $x=5$ and $y=4$ into the first part of the expression.

Substitute $x=5$ and $y=4$ into the second part of the expression.

Perform the division.

Let the other number be $N$ .

Set up the equation.

Solve for $N$ .

Perform the addition.

Find the remainder when 10 is divided by 4.

Determine the rate of loss per hour.

Multiply the rate by the total hours.

Find the total fraction of money spent.

Find a common denominator, which is 12.

Add the fractions.

Subtract the spent fraction from the total (1, or $\frac{12}{12}$ ).

List the universal set $\xi$ .

Find the intersection of sets A and B ( $A \cap B$ ), which are the elements common to both sets.

Find the complement of $(A \cap B)$ , denoted as $(A \cap B)'$ .

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More Mathematics Questions

Which of these years was a leap year: 2020, 2021, 2022?

Step-by-step solution

One million is 1,000,000.

Half a million is 500,000.

Add these values.

Expand 727^272.

Expand 535^353.

Multiply the expanded values.

A leap year is generally divisible by 4.

Check each year

Multiply the whole number by the denominator.

Add the numerator to the result.

Place this sum over the original denominator.

Identify the first four significant figures

Look at the fifth significant figure, which is 6.

Since 6 is 5 or greater, round up the fourth significant figure (3).

Substitute x=5x=5x=5 and y=4y=4y=4 into the first part of the expression.

Substitute x=5x=5x=5 and y=4y=4y=4 into the second part of the expression.

Perform the division.

Let the other number be NNN.

Set up the equation.

Solve for NNN.

Perform the addition.

Find the remainder when 10 is divided by 4.

Determine the rate of loss per hour.

Multiply the rate by the total hours.

Find the total fraction of money spent.

Find a common denominator, which is 12.

Add the fractions.

Subtract the spent fraction from the total (1, or 1212\frac{12}{12}1212​).

List the universal set ξ\xiξ.

Find the intersection of sets A and B (A∩BA \cap BA∩B), which are the elements common to both sets.

Find the complement of (A∩B)(A \cap B)(A∩B), denoted as (A∩B)′(A \cap B)'(A∩B)′.

Still stuck on this one?

More Mathematics Questions

Expand $7^2$ .

Expand $5^3$ .

Substitute $x=5$ and $y=4$ into the first part of the expression.

Substitute $x=5$ and $y=4$ into the second part of the expression.

Let the other number be $N$ .

Solve for $N$ .

Subtract the spent fraction from the total (1, or $\frac{12}{12}$ ).

List the universal set $\xi$ .

Find the intersection of sets A and B ( $A \cap B$ ), which are the elements common to both sets.

Find the complement of $(A \cap B)$ , denoted as $(A \cap B)'$ .