Question

using division to write fractions as decimals - item 52313
calculators cannot display repeating decimals. calculators always round.
yuris calculator displays decimals to 5 places. he uses his calculator to find the
decimal equivalent for each of the fractions below.
for each number, choose what his calculator will display as the decimal equivalent.
then choose whether or not this is the exact decimal equivalent.

1. \\(\frac{1}{9}\\)

calculator display for \\(\frac{1}{9}\\)
is this the exact decimal equivalent?

1. \\(\frac{7}{8}\\)

calculator display for \\(\frac{7}{8}\\)
is this the exact decimal equivalent?

1. \\(\frac{9}{16}\\)

calculator display for \\(\frac{9}{16}\\)
is this the exact decimal equivalent?

Explanation:

Step1: Analyze \(\frac{1}{9}\)

To find the decimal equivalent of \(\frac{1}{9}\), we perform the division: \(1\div9 = 0.11111\cdots\) (repeating decimal). Since the calculator displays up to 5 decimal places, it will show \(0.11111\). The exact decimal equivalent of \(\frac{1}{9}\) is a repeating decimal (\(0.\overline{1}\)), so the calculator's display (\(0.11111\)) is not the exact equivalent (because the repeating continues infinitely, and the calculator rounds or truncates after 5 places, but in this case, the first 5 digits are \(1\)s, but the exact value is non - terminating).

Step2: Analyze \(\frac{7}{8}\)

Calculate \(7\div8\): \(7\div8 = 0.875\). Since this is a terminating decimal, and the calculator can display up to 5 decimal places, it will show \(0.87500\) (or just \(0.875\) as trailing zeros after the non - zero decimal part don't change the value, but to 5 decimal places it's \(0.87500\)). And since \(7\div8 = 0.875\) exactly, this is the exact decimal equivalent.

Step3: Analyze \(\frac{9}{16}\)

Calculate \(9\div16\): \(9\div16=0.5625\). Since this is a terminating decimal, the calculator (displaying up to 5 decimal places) will show \(0.56250\) (or \(0.5625\) with a trailing zero to 5 decimal places). And since \(9\div16 = 0.5625\) exactly, this is the exact decimal equivalent.

For \(\frac{1}{9}\):

• Calculator display: \(0.11111\)
• Is this the exact decimal equivalent? No

For \(\frac{7}{8}\):

• Calculator display: \(0.87500\) (or \(0.875\))
• Is this the exact decimal equivalent? Yes

For \(\frac{9}{16}\):

• Calculator display: \(0.56250\) (or \(0.5625\))
• Is this the exact decimal equivalent? Yes

(Note: If we consider the calculator's display for \(\frac{7}{8}\) and \(\frac{9}{16}\) without the trailing zero (since \(0.875\) and \(0.5625\) are already terminating and the calculator may just show the non - zero part up to 5 places, but mathematically \(0.875 = 0.87500\) and \(0.5625=0.56250\) to 5 decimal places))

Answer:

s:

• For \(\frac{1}{9}\):
• Calculator display: \(0.11111\)
• Is exact? No
• For \(\frac{7}{8}\):
• Calculator display: \(0.87500\) (or \(0.875\))
• Is exact? Yes
• For \(\frac{9}{16}\):
• Calculator display: \(0.56250\) (or \(0.5625\))
• Is exact? Yes