Question

Change these fractions into decimals. Express each answer to three significant figures: (a) \(\frac{5}{6}\) (b) \(\frac{3}{7}\) (c) \(\frac{12}{16}\) (d) \(\frac{9}{18}\)

Short answer

(a) 0.833 (b) 0.429 (c) 0.750 (d) 0.500

Step-by-step solution

Identify the Fraction

First, identify the fractions given in the problem. They are \(\frac{5}{6}\), \(\frac{3}{7}\), \(\frac{12}{16}\), and \(\frac{9}{18}\).

Set Up Division

Convert each fraction to its decimal form by dividing the numerator by the denominator.

Perform Division for \(\frac{5}{6}\)

Divide 5 by 6. \[ \frac{5}{6} = 0.83333\text{...} \] Round to three significant figures: 0.833.

Perform Division for \(\frac{3}{7}\)

Divide 3 by 7. \[ \frac{3}{7} = 0.42857\text{...} \] Round to three significant figures: 0.429.

Perform Division for \(\frac{12}{16}\)

Divide 12 by 16. \[ \frac{12}{16} = 0.75 \] The result already has three significant figures: 0.750.

Perform Division for \(\frac{9}{18}\)

Divide 9 by 18. \[ \frac{9}{18} = 0.5 \] The result already has three significant figures: 0.500.

Key concepts

significant figures

Significant figures are the digits in a number that are important in representing its precision. These digits include all integers except leading zeros, which are just placeholders.
For instance, in the number 0.0056, there are two significant figures: 5 and 6. Similarly, in the number 123.45, there are five significant figures: 1, 2, 3, 4, and 5. It’s important to remember:

• All non-zero numbers are significant.
• Zeros within a number are significant.
• Leading zeros are not significant.
• Trailing zeros after a decimal point are significant.

So, when the exercise specifies rounding to three significant figures, we keep only the first three significant numbers, adjusting the last one based on the subsequent digit.

fraction division

Fraction division is a straightforward process that involves dividing the numerator by the denominator. This process converts a fraction to its decimal form.
For example, consider \[ \frac{5}{6} \](5/6). Here, 5 is the numerator and 6 is the denominator. We divide 5 by 6 to get the decimal 0.83333…
It’s the same process for any fraction; simply perform the division:

• \[ \frac{3}{7} = 0.42857… \]
• \[ \frac{12}{16} = 0.75 \]
• \[ \frac{9}{18} = 0.5 \]

This conversion is essential for expressing fractions in decimal form and will help when rounding to a specific number of significant figures.

rounding numbers

Rounding numbers is the process of reducing the digits in a number while keeping its value close to what it was. This is useful when we need to express a number in a more manageable form, especially when we're working with significant figures.
To round a number to a certain number of significant figures, follow these steps:

• Identify the number of significant digits required.
• Look at the digit immediately after the last significant figure.
• If this digit is 5 or more, round up the last significant figure.
• If this digit is less than 5, keep the last significant figure as it is.

For example, rounding \[ \frac{5}{6} = 0.83333… \] to three significant figures gives us 0.833. Here are some more examples:

• \[ \frac{3}{7} = 0.42857… \] rounded to 0.429
• \[ \frac{12}{16} = 0.75 \] is already 0.750 with three significant figures
• \[ \frac{9}{18} = 0.5 \] becomes 0.500 with three significant figures

Rounding helps in making complex calculations simpler and the results easy to understand.