(a) Write these numbers in order of size, starting with the smallest. [2] \(\frac{13}{20}\) \(0.66\) \(64.5\%\) \(\frac{2}{3}\) (b) Write \(\frac{2}{3}\) as...

(a) The four numbers are given as fractions, a decimal and a percentage, so convert them all to decimals before comparing.

• \( \frac{13}{20} = 13 \div 20 = 0.65 \)
• \( 0.66 \) is already a decimal
• \( 64.5\% = 64.5 \div 100 = 0.645 \)
• \( \frac{2}{3} = 2 \div 3 = 0.6666\ldots \)

Writing each number as a decimal is the method step [M1]

Padding to three decimal places gives \( 0.650 \), \( 0.660 \), \( 0.645 \) and \( 0.666\ldots \), so the order from smallest is \( 0.645 \lt 0.650 \lt 0.660 \lt 0.666\ldots \).

Back in the original forms, starting with the smallest: \( 64.5\% \), \( \frac{13}{20} \), \( 0.66 \), \( \frac{2}{3} \) [A1]

(b) \( \frac{2}{3} = 0.6666\ldots \). The first three significant figures are \( 6, 6, 6 \) and the next digit is \( 6 \), which is \( 5 \) or more, so the last one rounds up: \( 0.667 \) [B1]

(c) From the conversions, \( \frac{13}{20} = 0.65 \) and \( 0.65 \lt 0.66 \), so

\[ \frac{13}{20} \lt 0.66 \] [B1]

A frequent error in part (b) is writing \( 0.666 \) by truncating the recurring decimal instead of rounding it. Rounding looks at the first digit discarded, and here that digit is a \( 6 \), so the final \( 6 \) becomes a \( 7 \). A second error is converting \( 64.5\% \) to \( 0.645 \) but then reading it as larger than \( 0.65 \) because it has more digits; compare \( 0.645 \) with \( 0.650 \) at equal length.