Problem 53 - Algebra A Combined Function

Question

Write each fraction as a decimal. If the decimal is a repeating decimal, write using the bar notation and then round to the nearest hundredth. $$ \frac{7}{16} $$

Step-by-Step Solution

Answer

0.44

Step 1: Understanding the Fraction

The problem requires converting the fraction \( \frac{7}{16} \) into a decimal. This involves performing division, with 7 being the dividend and 16 the divisor.

Step 2: Perform Long Division

Divide 7 by 16. Since 16 does not go into 7, consider 70 (by adding a decimal and a zero). 16 goes into 70 four times (as 16 x 4 = 64) with a remainder of 6. Bring down another zero to the remainder to make it 60, then 16 into 60 goes three times (as 16 x 3 = 48) with a remainder of 12. Continue this process until a pattern emerges or there is no remainder.

Step 3: Determine Decimal Representation

Continuing the division, 16 goes into 120 seven times (as 16 x 7 = 112) with a remainder of 8. Bring down another zero to have 80, and 16 goes into 80 five times (as 16 x 5 = 80) exactly, with no remainder. Thus, the division ends here.

Step 4: Identify the Resulting Decimal

From the long division process, the complete decimal of \( \frac{7}{16} \) is 0.4375.

Step 5: Round to the Nearest Hundredth

To round 0.4375 to the nearest hundredth, look at the third digit (7). This digit is greater than 5, so we round up the second digit, resulting in 0.44.

Key Concepts

Long Division

The process of transforming a fraction like \( \frac{7}{16} \) into a decimal begins with long division. In this method, we divide the numerator (7) by the denominator (16). Since 16 cannot divide into 7, we write a decimal point followed by a zero, making it 70. This step allows us to see how many times 16 fits into 70. The fit is four times (because 16 x 4 = 64), which gives a remainder of 6.
Here's how long division works step by step:

• Add '0' to the dividend to shift the decimal.
• Determine how many times the divisor fits into the dividend.
• Subtract the result from the current number (e.g., 70 - 64 = 6).
• Bring down the next '0' and repeat the process (thus making 60, and then 120, etc.).

Repeat this method until the remainder is zero or a repeating pattern begins. In our case, this process results in an exact decimal representation.

Decimal Representation

Decimal representation of a fraction offers an alternative view. After completing the long division, you obtain a series of digits as the decimal value. For our fraction, \( \frac{7}{16} \), the result of long division is 0.4375. Each digit after the decimal point represents a fraction of a ten, adding positional value to the decimal.
Decimal representation is useful because it makes comparison and arithmetic with fractional numbers straightforward. When you write a fraction as a decimal, you essentially represent the same value in an entirely different base system (base 10). In this example, 0.4375 precisely equals \( \frac{7}{16} \) with no fractions lost or ignored.
Understanding decimals helps in a wide array of real-world applications, like calculating measurements or finances, where exact fractions might be inconvenient or cumbersome.

Rounding Decimals

Rounding decimals is a crucial skill for simplifying values, especially when working with lengthy decimal representations. To round a decimal, we look at a particular digit place that we are rounding to, and then examine the digit that follows it.
In rounding the decimal 0.4375 to the nearest hundredth, you:

• Identify the place you need to round to—here, the hundredths place is 3.
• Check the digit immediately after it, which is the thousandths place (7).
• If the digit is 5 or higher, add one to the hundredths place. If it's lower, you leave the hundredths in place.

Hence, since the digit in the thousandths place of 0.4375 is 7, we round the hundredths place up, resulting in 0.44.
Rounding is helpful for simplifying numbers for everyday use. It helps ensure numbers are more readable and easier to communicate, especially in scenarios where extreme precision isn't necessary.