Show that each number is a rational number by writing it as the quotient of two integers. 2/15
To show that a number is a rational number, we need to express it as the quotient of two integers.
In this case, we have the number 2/15. To express this as a quotient of two integers, we set the numerator (the top number) as the dividend and the denominator (the bottom number) as the divisor. We can write it as:
2/15 = 2 ÷ 15
Now, let's perform the division.
When we divide 2 by 15, we get the quotient 0.1333... (the decimal repeats infinitely). However, to express it as a quotient of two integers, we need to find whole numbers that can be divided to result in this decimal.
To convert this repeating decimal to a fraction, we use the following method:
Let x = 0.1333...
Multiplying both sides of the equation by 10000 to move the decimal places:
10000x = 1333.333...
Now, subtract the original equation from this one:
10000x - x = 1333.333... - 0.1333...
Simplifying:
9999x = 1333.2
To continue, we can approximate the decimal value manually or using a calculator, and then simplify the fraction.
Rounding the decimal to a few decimal places, we have:
9999x = 1333.2
x ≈ 1333.2 / 9999
x ≈ 0.1332133213...
Since the decimal 0.1332133213... is repeating, we can write it as a fraction:
x = 0.1332133213... ≈ 1332/9999
Therefore, we have shown that 2/15 is a rational number since it can be expressed as the quotient 1332/9999, where both the numerator and denominator are integers.