OMPT-C mixed paper

Answer: $8$

$24 = 8 \times 3$ and $88 = 8 \times 11\mathclose{,}$ and $3$ and $11$ have no common factor left, so the greatest common divisor is $8\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the least common multiple instead → $264$

Answer: $-25$

Division and multiplication have the same priority, so work from left to right: $-15 \div (-3) = 5\mathclose{,}$ then $5 \times (-5) = -25\mathclose{.}$ (An odd number of minus signs, so the answer is negative.)

Common mistakes, and the answer each one gives:

• Got the sign of the answer wrong → $25$
• Multiplied first, then divided → $-1$

Answer: $\dfrac{187}{30}$

Write all three over the lowest common denominator, $30\mathclose{,}$ then combine the numerators: $\dfrac{7}{5} + \dfrac{7}{3} + \dfrac{5}{2} = \dfrac{42}{30} + \dfrac{70}{30} + \dfrac{75}{30} = \dfrac{187}{30}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added (or subtracted) the numerators and the denominators separately → $\frac{19}{10}$
• Wrote all three over the common denominator without changing the numerators → $\frac{19}{30}$

Answer: $\dfrac{22}{45}$

Write both fractions over the lowest common denominator, then combine the numerators: $\dfrac{3}{5} - \dfrac{1}{9} = \dfrac{27}{45} - \dfrac{5}{45} = \dfrac{22}{45}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added (or subtracted) the numerators and the denominators separately → $- \frac{1}{2}$
• Wrote both over $45$ but did not change the numerators → $\frac{2}{45}$

Answer: $0.4$

Divide the numerator by the denominator: $2 \div 5 = 0.4\mathclose{.}$

Common mistakes, and the answer each one gives:

• Wrote the numerator and denominator as the digits after the point → $0.25$
• Divided the denominator by the numerator → $2.5$

Answer: $\dfrac{4}{5}$

Take the square root of the numerator and of the denominator: $\sqrt{\dfrac{16}{25}} = \dfrac{\sqrt{16}}{\sqrt{25}} = \dfrac{4}{5}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Halved the fraction instead of taking its square root → $\frac{8}{25}$
• Took the square root of the numerator only → $\frac{4}{25}$

Answer: $-2$

Work out each inside first, then make it non-negative: $-1 - 3 = -4\mathclose{,}$ so the first bar gives $4\mathclose{;}$ $-5 - 1 = -6\mathclose{,}$ so the second gives $6\mathclose{.}$ Then $4 - 6 = -2\mathclose{.}$

Common mistakes, and the answer each one gives:

• Dropped the bars without making the insides positive → $2$
• Applied the bars to each number on its own, as if $|a - b| = |a| - |b|$ → $-6$

Answer: $\dfrac{109}{72}$

Multiplication comes before addition, so work out $\dfrac{7}{9} \times \dfrac{7}{8} = \dfrac{49}{72}$ first, then add $\dfrac{5}{6}\mathclose{:}$ $\dfrac{109}{72}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{203}{144}$

Answer: $67$

Put each value in BRACKETS, so the minus sign is squared too: $5\left(-3\right)^{2} - 3\left(-3\right)\left(2\right) + 2\left(2\right) = 45 + 18 + 4 = 67\mathclose{.}$

Common mistakes, and the answer each one gives:

• Squared the number but not its minus sign, so $a^2$ came out negative → $-23$
• Doubled $a$ instead of squaring it → $-8$

Answer: $x^{\frac{11}{2}}$

$\sqrt{x} = x^{\frac{1}{2}}\mathclose{,}$ and dividing by $x^{-3}$ subtracts $-3\mathclose{,}$ which ADDS $3\mathclose{:}$ $2 + \frac{1}{2} + 3 = \frac{11}{2}\mathclose{,}$ so the answer is $x^{\frac{11}{2}}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Treated $\sqrt{x}$ as $x^2$ → $x^{7}$
• Subtracted the negative exponent as if it were positive → $\frac{1}{\sqrt{x}}$

Answer: $\dfrac{8\sqrt{3}}{3}$

Multiply top and bottom by $\sqrt{3}\mathclose{,}$ since $\sqrt{3} \cdot \sqrt{3} = 3\mathclose{:}$ $\dfrac{8\sqrt{3}}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied only the denominator by the root → $\dfrac{8}{3}$
• Moved the root to the top without dividing by it → $8\sqrt{3}$

Answer: $9 x^{2} - 54 x + 81$

$(p - q)^2 = p^2 - 2pq + q^2\mathclose{,}$ so $\left(3 x - 9\right)^{2} = \left(3x\right)^{2} - 2 \cdot 3x \cdot 9 + 9^{2} = 9 x^{2} - 54 x + 81\mathclose{.}$

Common mistakes, and the answer each one gives:

• Squared each term and left out the middle term → $9 x^{2} + 81$
• Gave the middle term the wrong sign → $9 x^{2} + 54 x + 81$
• Forgot to double the middle term → $9 x^{2} - 27 x + 81$

Answer: $3\left(x - 4\right)\left(x + 4\right)$

Take out the common factor $3$ first: $3 x^{2} - 48 = 3\left(x^{2} - 16\right)\mathclose{.}$ Then $x^{2} - 16 = x^{2} - 4^{2}$ is a difference of two squares, $p^2 - q^2 = (p - q)(p + q)\mathclose{,}$ so the answer is $3\left(x - 4\right)\left(x + 4\right)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Wrote it as a perfect square → $3\left(x - 4\right)^{2}$
• Left out the common factor $3$ → $\left(x - 4\right)\left(x + 4\right)$

Answer: $\dfrac{x + 7}{x - 1}$

Factorise both: $\dfrac{\left(x + 1\right) \left(x + 7\right)}{\left(x - 1\right) \left(x + 1\right)}\mathclose{.}$ The factor $x + 1$ is common to both and cancels, leaving $\dfrac{x + 7}{x - 1}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Cancelled the wrong bracket, keeping the common one → $\frac{x + 1}{x - 1}$
• Cancelled the $x^2$ terms, which are terms and not factors → $- 8 x - 7$

Answer: $\dfrac{2 \left(3 x - 2\right)}{\left(x + 6\right)\left(x + 1\right)}$

The common denominator is $(x + 6)(x + 1)\mathclose{,}$ so multiply each numerator by the OTHER bracket: $\dfrac{8(x + 1) - 2(x + 6)}{\left(x + 6\right)\left(x + 1\right)} = \dfrac{6 x - 4}{\left(x + 6\right)\left(x + 1\right)} = \dfrac{2 \left(3 x - 2\right)}{\left(x + 6\right)\left(x + 1\right)}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Subtracted the numerators and the denominators separately → $\dfrac{6}{5}$
• Did not take the minus sign through the whole of $2(x + 6)$ → $\dfrac{6 x + 20}{\left(x + 6\right)\left(x + 1\right)}$
• Multiplied each numerator by its OWN denominator instead of the other one → $\dfrac{6 x + 46}{\left(x + 6\right)\left(x + 1\right)}$

Answer: $3$

Multiply both sides by $\left(x - 4\right)\left(x - 5\right)\mathclose{:}$ $1 - x = A\left(x - 5\right) + B\left(x - 4\right)\mathclose{.}$ Put $x = 4$ to lose the $B$ term: $-3 = A \cdot (-1)\mathclose{,}$ so $A = 3\mathclose{.}$ Put $x = 5\mathclose{:}$ $-4 = B \cdot 1\mathclose{,}$ so $B = -4\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave B instead → $-4$
• Substituted $x = -4$ instead of $x = 4$ → $-\dfrac{5}{9}$

Answer: $t = \dfrac{2s}{u + v}$

Undo what is done to $t$ in reverse order, the same to both sides: multiply both sides by 2, $2s = (u + v)t\mathclose{,}$ then divide by the WHOLE bracket $u + v\mathclose{.}$ So $t = \dfrac{2s}{u + v}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Halved instead of doubling → $t = \dfrac{s}{2(u + v)}$
• Divided by $u$ and by $v$ separately and added → $t = \dfrac{2s}{u} + \dfrac{2s}{v}$

Answer: $4$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{4 - 0}{-2 - (-3)} = 4\mathclose{.}$

Common mistakes, and the answer each one gives:

• Inverted the fraction (run over rise) → $\frac{1}{4}$
• Subtracted the coordinates in opposite orders → $-4$

Answer: $(-3, -5)$

Both give the same $y$ there: $3 x + 4 = x - 2\mathclose{,}$ so $2 x = -6$ and $x = -3\mathclose{.}$ Then $y = -5\mathclose{:}$ the point is $(-3, -5)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Moved the $x$ term across without changing its sign → $(-\dfrac{3}{2}, -\dfrac{1}{2})$
• Found $x$ and gave it as both coordinates → $(-3, -3)$

Answer: $\dfrac{25}{2}$

Expand the bracket: $5 x + 30 = 7 x + 5\mathclose{.}$ Put the $x$ terms on the left and the numbers on the right, changing the sign of each term that crosses over: $- 2 x = -25\mathclose{,}$ so $x = \dfrac{25}{2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $\frac{1}{2}$
• Moved the number across without changing its sign → $- \frac{35}{2}$

Answer: $\dfrac{40}{3}$

Multiply both sides by $28$ to clear the denominators: $4\left(x - 4\right) = 7\left(x - 8\right)\mathclose{.}$ Expand: $4 x - 16 = 7 x - 56\mathclose{.}$ Collect: $- 3 x = -40\mathclose{,}$ so $x = \dfrac{40}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied out $7\left(x - 8\right)$ as $7 x - 8\mathclose{,}$ multiplying only the $x$ → $- \frac{8}{3}$
• Cross-multiplied the wrong way, writing $7\left(x - 4\right) = 4\left(x - 8\right)$ → $- \frac{4}{3}$

Answer: $x > -2$

Collect the $x$ terms on the left and the numbers on the right: $3 x > -6\mathclose{.}$ Divide by $3\mathclose{,}$ a positive number, so the sign stays: $x > -2\mathclose{.}$

Common mistakes, and the answer each one gives:

• Reversed the inequality because the right-hand side was negative → $x < -2$
• Moved the number across without changing its sign → $x > - \frac{8}{3}$

Answer: $x > -1$

Collect the $x$ terms on the left and the numbers on the right: $- 4 x < 4\mathclose{.}$ Divide by $-4$ — a NEGATIVE number, so the inequality sign turns round: $x > -1\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not reverse the inequality when dividing by a negative number → $x < -1$
• Moved the number across without changing its sign → $x > \frac{3}{2}$

Answer: $82$

Vertically opposite angles are equal, so it is also $82^\circ\mathclose{.}$ (The two angles NEXT to it are $98^\circ$ each.)

Common mistakes, and the answer each one gives:

• Gave an angle next to it instead → $98$

Answer: $156$

Angles round a point add to $360^\circ\mathclose{:}$ $360 - 134 - 70 = 156\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took only $134^\circ$ away from $360^\circ$ → $226$

Answer: $75$

The other two angles are equal and share what is left of $180^\circ\mathclose{:}$ $\dfrac{180 - 30}{2} = 75\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not halve what was left → $150$
• Halved the top angle instead → $15$

Answer: $46$

The other two angles are equal and share what is left of $180^\circ\mathclose{:}$ $\dfrac{180 - 88}{2} = 46\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not halve what was left → $92$
• Halved the top angle instead → $44$

Answer: $24$

Rearrange Pythagoras: the missing leg squared is $25^2 - 7^2 = 576\mathclose{,}$ so the leg is $\sqrt{576} = 24\mathclose{.}$

Common mistakes, and the answer each one gives:

• Subtracted the lengths instead of their squares → $18$
• Added the squares, which is the rule for the HYPOTENUSE → $\sqrt{674}$

Answer: $21$

Rearrange Pythagoras: the missing leg squared is $29^2 - 20^2 = 441\mathclose{,}$ so the leg is $\sqrt{441} = 21\mathclose{.}$

Common mistakes, and the answer each one gives:

• Subtracted the lengths instead of their squares → $9$
• Added the squares, which is the rule for the HYPOTENUSE → $\sqrt{1241}$

Answer: $10$

Pythagoras: the square on the hypotenuse equals the sum of the squares on the legs. $6^2 + 8^2 = 100\mathclose{,}$ and $\sqrt{100} = 10\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added the legs instead of their squares → $14$
• Subtracted the squares, which finds a LEG, not the hypotenuse → $2 \sqrt{7}$