(a)
Write down the fraction of the rectangle that is shaded.
Give your answer in its simplest form. [2]
(b) Write down a fraction that is equivalent to \(\frac{7}{12}\). [1]
(c) Write down a fraction that completes this calculation. [1]
(d) Find a fraction that makes this statement true. [1]
(e) Write these numbers in order, starting with the smallest. [2]
\(5.7 \times 10^{-1}\) \(\frac{4}{7}\) \(\sqrt{0.33}\) 57.2%
▶️ Answer/Explanation
(a) Ans: \(\frac{4}{15}\)
Total squares = 15, Shaded squares = 4. Fraction = \(\frac{4}{15}\) (simplest form).
(b) Ans: \(\frac{14}{24}\) (or any equivalent)
Multiply numerator and denominator by the same integer (e.g., 2 gives \(\frac{14}{24}\)).
(c) Ans: \(\frac{11}{13}\)
Since \(\frac{3}{13} + \frac{8}{13} = \frac{11}{13}\), the missing fraction is \(\frac{11}{13}\).
(d) Ans: Any fraction > \(\frac{3}{8}\)
Example: \(\frac{1}{2}\) because \(\frac{3}{8} < \frac{1}{2}\).
(e) Ans: \(5.7 \times 10^{-1}, \frac{4}{7}, 57.2\%, \sqrt{0.33}\)
Convert all to decimals: \(0.57, 0.571…, 0.572, 0.574\). Thus, the order is \(0.57 < 0.571… < 0.572 < 0.574\).
(a) Garcia and Elena are each given \( x \) dollars.
(i) Elena spends 4 dollars.
Write down an expression in terms of \( x \) for the number of dollars she has now.
(ii) Garcia doubles his money by working and then is given another 5 dollars.
Write down an expression in terms of \( x \) for the number of dollars he has now.
(iii) Garcia now has three times as much money as Elena.
Write down an equation in \( x \) to show this.
(iv) Solve the equation to find the value of \( x \).
(b) Solve the simultaneous equations
\( \begin{cases} 3x – 2y = 3, \\ x + 4y = 8. \end{cases} \)
▶️ Answer/Explanation
(a)(i) Ans: \( x – 4 \)
Elena starts with \( x \) dollars and spends 4, so she has \( x – 4 \) dollars left.
(a)(ii) Ans: \( 2x + 5 \)
Garcia doubles his money to \( 2x \) dollars and then receives 5 more, totaling \( 2x + 5 \).
(a)(iii) Ans: \( 2x + 5 = 3(x – 4) \)
Garcia’s amount (\( 2x + 5 \)) is three times Elena’s amount (\( x – 4 \)).
(a)(iv) Ans: \( x = 17 \)
Solve the equation: \( 2x + 5 = 3x – 12 \). Subtract \( 2x \): \( 5 = x – 12 \). Add 12: \( x = 17 \).
(b) Ans: \( x = 2 \), \( y = 1.5 \)
Multiply the first equation by 2: \( 6x – 4y = 6 \).
Add the second equation: \( (6x – 4y) + (x + 4y) = 6 + 8 \), simplifying to \( 7x = 14 \), so \( x = 2 \).
Substitute \( x = 2 \) into \( x + 4y = 8 \): \( 2 + 4y = 8 \), \( 4y = 6 \), \( y = 1.5 \).