Question 1

What is a rational number?

Accepted Answer

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, where the numerator p is an integer and the denominator q is a non-zero integer. Examples include 1/2, -3/4, 5 (which is 5/1), and 0.75 (which is 3/4). All integers and terminating or repeating decimals are rational numbers.

Question 2

How do you add or subtract rational numbers?

Accepted Answer

To add or subtract rational numbers (fractions): 1) Find the least common denominator (LCD) of both fractions, 2) Convert each fraction to an equivalent fraction with the LCD, 3) Add or subtract the numerators while keeping the common denominator, 4) Simplify the result by dividing both numerator and denominator by their greatest common divisor (GCD).

Question 3

How do you multiply rational numbers?

Accepted Answer

To multiply rational numbers, simply multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. For example, (a/b) × (c/d) = (a×c)/(b×d). After multiplication, simplify the result by finding the GCD of the numerator and denominator and dividing both by it.

Question 4

How do you divide rational numbers?

Accepted Answer

To divide rational numbers, multiply the first fraction by the reciprocal of the second fraction. The reciprocal is obtained by flipping the numerator and denominator. For example, (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c). Then simplify the result.

Question 5

What is the least common denominator (LCD)?

Accepted Answer

The least common denominator (LCD) is the smallest positive integer that is divisible by all denominators in a set of fractions. It's the least common multiple (LCM) of the denominators. Finding the LCD is essential for adding or subtracting fractions, as it allows you to convert fractions to equivalent forms with the same denominator.

Question 6

How do you simplify a rational number?

Accepted Answer

To simplify a rational number (fraction), find the greatest common divisor (GCD) of the numerator and denominator, then divide both by this GCD. For example, to simplify 12/18, find GCD(12, 18) = 6, then divide: 12÷6 = 2 and 18÷6 = 3, resulting in 2/3. A fraction is fully simplified when the GCD of numerator and denominator is 1.

Question 7

What is a mixed number?

Accepted Answer

A mixed number is a combination of a whole number and a proper fraction, written side by side. For example, 2 1/3 means 2 + 1/3. Mixed numbers are another way to express improper fractions (where the numerator is larger than the denominator). To convert an improper fraction to a mixed number, divide the numerator by the denominator; the quotient is the whole number, and the remainder becomes the new numerator.

Question 8

Are all decimals rational numbers?

Accepted Answer

Not all decimals are rational numbers. Terminating decimals (like 0.5 = 1/2) and repeating decimals (like 0.333... = 1/3) are rational because they can be expressed as fractions. However, non-terminating, non-repeating decimals like π (3.14159...) and √2 (1.41421...) are irrational numbers that cannot be expressed as fractions of integers.

Question 9

How do you convert a decimal to a rational number?

Accepted Answer

To convert a terminating decimal to a rational number: 1) Count the decimal places (n), 2) Write the decimal as a fraction with the decimal number as numerator and 10^n as denominator, 3) Simplify. For example, 0.75 has 2 decimal places, so 0.75 = 75/100 = 3/4. For repeating decimals, the process involves algebraic manipulation to find the fractional form.

Question 10

What properties do rational numbers have?

Accepted Answer

Rational numbers have several important properties: 1) Closure: The sum, difference, product, and quotient (except division by zero) of rational numbers is rational, 2) Commutative: a+b = b+a and a×b = b×a, 3) Associative: (a+b)+c = a+(b+c) and (a×b)×c = a×(b×c), 4) Distributive: a×(b+c) = a×b + a×c, 5) Density: Between any two rational numbers, there exists another rational number.

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