Work out the value of \(xy\). Give your answer as a fraction in its simplest form. (5)
Mark scheme
Answer
Mark
Mark scheme
\(\dfrac{5}{33}\)
M1
for \(10x = 2.\dot{2}\) or 2.22… or \((10x - x =)\ 2.\dot{2} - 0.\dot{2}\ (= 2)\) or \(2.22\ldots - 0.22\ldots\ (= 2)\) or \(\dfrac{2}{9}\) oe fraction
M1
for a method using two recurring decimals that leads to a terminating decimal difference, using correct multiples of \(y\) eg \((1000y - 10y =)\ 681.\dot{8}\dot{1} - 6.\dot{8}\dot{1}\ (= 675)\) or \(681.81\ldots - 6.81\ldots\ (= 675)\) or \(\dfrac{675}{990}\) or \((100y - y) = 68.\dot{1}\dot{8} - 0.6\dot{8}\dot{1}\ (= 67.5)\) or \(68.181\ldots - 0.681\ldots\ (= 67.5)\) or \(\dfrac{67.5}{99}\)
A1
for \((x =)\ \dfrac{2}{9}\) oe and \((y =)\ \dfrac{675}{990}\) oe
M1
for \(\text{``}\dfrac{2}{9}\text{''} \times \text{``}\dfrac{675}{990}\text{''}\)
A1
cao
Additional guidance
eg \(\dfrac{20}{90}, \dfrac{22}{99}\)
Accept \((y =)\ \dfrac{67.5}{99}\)
Award 4 marks for an answer equivalent to \(\dfrac{5}{33}\), eg \(\dfrac{15}{99}, \dfrac{135}{891}, \dfrac{1350}{8910}\) unless from incorrect working
14 Prove algebraically that \(0.4\dot{6}\dot{2}\) can be written as \(\dfrac{229}{495}\) (3)
Mark scheme
Answer
Mark
Mark scheme
Shown
M1
for start to find multiples of \(x\) with the same recurring pattern eg for \((10x =)\ 4.626262\ldots\) or \((100x =)\ 46.262626\ldots\) or \((1000x =)\ 462.626262\ldots\)
M1
(dep on M1) for a correct subtraction that would lead to a terminating decimal, eg \((1000x - 10x) = 462.6262\ldots - 4.6262\ldots\ (= 458)\) or \((100x - x) = 46.2626\ldots - 0.4626\ldots\ (= 45.8)\)
C1
for correct working leading to the correct answer
Additional guidance
Any recurring notation acceptable throughout. Proofs with terminating decimals (at least 5 figures) score M1M1C0
