Link decimal arithmetic to exact fractions and place value.
Core & Extended · Syllabus C1.4, C1.6, E1.4, E1.6
Notes and key methods
For addition and subtraction, align decimal points so equal place values meet. For multiplication, one can multiply the corresponding integers and restore the total number of decimal places. Dividing both numbers by the same nonzero scale does not change a quotient.
A terminating decimal is an exact fraction with a power of ten as denominator. Converting to that fraction is often the clearest non-calculator check. A small positive divisor can make a quotient larger, not smaller.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Decimals and place value
Let \(\displaystyle A=3.9\) and \(\displaystyle B=0.05\). Treat the displayed decimals as exact.
Calculate A + B.
Calculate A − B.
Calculate A × B.
Calculate A ÷ B.
Show the reasoning
(a)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{79}{20}\)
Answer: \(\displaystyle \frac{79}{20}\).
(b)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{77}{20}\)
Answer: \(\displaystyle \frac{77}{20}\).
(c)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{39}{200}\)
Answer: \(\displaystyle \frac{39}{200}\).
(d)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(78\)
Answer: \(78\).
Example 2 · Decimals and place value
Let \(\displaystyle A=4.6\) and \(\displaystyle B=0.06\). Treat the displayed decimals as exact.
Calculate A + B.
Calculate A − B.
Calculate A × B.
Calculate A ÷ B.
Show the reasoning
(a)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{233}{50}\)
Answer: \(\displaystyle \frac{233}{50}\).
(b)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{227}{50}\)
Answer: \(\displaystyle \frac{227}{50}\).
(c)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{69}{250}\)
Answer: \(\displaystyle \frac{69}{250}\).
(d)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{230}{3}\)
Answer: \(\displaystyle \frac{230}{3}\).
Example 3 · Decimals and place value
Let \(\displaystyle A=5.3\) and \(\displaystyle B=0.07\). Treat the displayed decimals as exact.
Calculate A + B.
Calculate A − B.
Calculate A × B.
Calculate A ÷ B.
Show the reasoning
(a)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{537}{100}\)
Answer: \(\displaystyle \frac{537}{100}\).
(b)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{523}{100}\)
Answer: \(\displaystyle \frac{523}{100}\).
(c)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{371}{1000}\)
Answer: \(\displaystyle \frac{371}{1000}\).
(d)
Replace the exact decimals by fractions with powers of ten, perform the operation and simplify.
\(\displaystyle \frac{530}{7}\)
Answer: \(\displaystyle \frac{530}{7}\).
Common mistakes
Keep intermediate values exact until the final step.