Express the number as a ratio of integers.
The Correct Answer and Explanation is:
Sure! Let’s say the number is:
0.16̅ (which means 0.161616… repeating)
Answer:
The number 0.16̅ can be expressed as the ratio of the integers 16/99.
Explanation
To express a repeating decimal like 0.161616… as a ratio of integers, we use algebraic steps to convert it into a fraction.
Let’s begin by letting x = 0.161616…
Because the repeating part (the block of digits that repeats) is 16, which is two digits long, we multiply both sides of the equation by 100. This shifts the decimal point two places to the right, aligning the repeating parts.
So,
100x = 16.161616…
Now subtract the original equation (x = 0.161616…) from this new equation:
100x = 16.161616…
– x = 0.161616…
99x = 16
Now solve for x by dividing both sides of the equation by 99:
x = 16/99
So, 0.161616… = 16/99, which is a ratio of two integers. This confirms that the decimal is a rational number.
This process works for any repeating decimal. The key idea is to eliminate the repeating part by multiplying the number by a power of 10 that matches the length of the repeating cycle. In this case, the repeating block is two digits long, so we use 100. If the repeating block had only one digit, like in 0.777…, we would use 10 instead.
By subtracting the original number from the shifted version, the repeating decimals cancel out. What remains is a simple equation that can be solved to give the fraction form.
This technique is a standard algebraic method used in middle and high school math to show that repeating decimals always represent rational numbers.
