Numbers

Answer: $\dfrac{25}{33}$

Write both fractions over the lowest common denominator, then combine the numerators: $\dfrac{1}{11} + \dfrac{2}{3} = \dfrac{3}{33} + \dfrac{22}{33} = \dfrac{25}{33}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added (or subtracted) the numerators and the denominators separately → $\frac{3}{14}$
• Wrote both over $33$ but did not change the numerators → $\frac{1}{11}$

Answer: $8$

The denominator was multiplied by $4$ $(5 \times 4 = 20)\mathclose{,}$ so the numerator must be too: $2 \times 4 = 8\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added to the top what was added to the bottom → $17$

Answer: $5$

$30 = 5 \times 6$ and $55 = 5 \times 11\mathclose{,}$ and $6$ and $11$ have no common factor left, so the greatest common divisor is $5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the least common multiple instead → $330$

Answer: $\dfrac{256}{625}$

Raise the numerator and the denominator to the power $4\mathclose{:}$ $4^{4} = 256$ and $5^{4} = 625\mathclose{.}$ An even power of a negative number is positive, so the answer is $\dfrac{256}{625}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Kept the minus sign: an even power of a negative number is positive → $- \frac{256}{625}$
• Multiplied by the exponent instead of raising to it → $- \frac{16}{5}$

Answer: $-3$

Work out each inside first, then make it non-negative: $-1 - 4 = -5\mathclose{,}$ so the first bar gives $5\mathclose{;}$ $0 + 8 = 8\mathclose{,}$ so the second gives $8\mathclose{.}$ Then $5 - 8 = -3\mathclose{.}$

Common mistakes, and the answer each one gives:

• Dropped the bars without making the insides positive → $-13$
• Applied the bars to each number on its own, as if $|a - b| = |a| - |b|$ → $-11$

Answer: $\dfrac{71}{32}$

Powers first, then multiplication, then subtraction: $\left(\dfrac{3}{4}\right)^{2} = \dfrac{9}{16}\mathclose{,}$ then $\dfrac{9}{16} \times \dfrac{1}{2} = \dfrac{9}{32}\mathclose{,}$ and $\dfrac{5}{2} - \dfrac{9}{32} = \dfrac{71}{32}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Worked strictly left to right, subtracting before multiplying → $\frac{31}{32}$
• Squared only the numerator → $\frac{11}{8}$

Answer: $\dfrac{15}{28}$

Dividing by a fraction is multiplying by it turned over: $\dfrac{3}{7} \div \dfrac{4}{5} = \dfrac{3}{7} \times \dfrac{5}{4} = \dfrac{15}{28}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied without turning the second fraction over → $\frac{12}{35}$
• Turned the FIRST fraction over instead of the second → $\frac{28}{15}$

Answer: $0.27$

Multiply as whole numbers, $3 \times 9 = 27\mathclose{,}$ then count the decimal places of BOTH factors — $1 + 1 = 2$ — so the answer is $0.27\mathclose{.}$

Common mistakes, and the answer each one gives:

• Counted the decimal places of only one factor → $2.7$

Answer: $84$

Their greatest common divisor is $4\mathclose{,}$ and $\text{lcm} = \dfrac{12 \times 28}{4} = 84\mathclose{:}$ the smallest number both divide into.

Common mistakes, and the answer each one gives:

• Gave the greatest common divisor instead → $4$
• Multiplied the two numbers → $336$

Answer: $\dfrac{20}{77}$

Multiply the numerators together and the denominators together, then cancel: $\dfrac{5}{11} \times \dfrac{4}{7} = \dfrac{20}{77}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Turned the second fraction over, as if dividing → $\frac{35}{44}$
• Multiplied the numerators but added the denominators → $\frac{10}{9}$

Answer: $\dfrac{3}{7}$

The greatest common factor of $54$ and $126$ is $18\mathclose{:}$ $\dfrac{54 \div 18}{126 \div 18} = \dfrac{3}{7}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Subtracted the common factor instead of dividing by it → $\dfrac{1}{3}$
• Divided only the numerator by the common factor → $\dfrac{1}{42}$

Answer: $\dfrac{27}{343}$

A negative exponent turns the fraction over, and the exponent $\frac{3}{2}$ means the square root raised to the power $3\mathclose{:}$ $\left(\dfrac{9}{49}\right)^{\frac{3}{2}} = \left(\dfrac{3}{7}\right)^{3} = \dfrac{27}{343}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{343}{27}$
• Took the root but never applied the power → $\frac{3}{7}$
• Turned the fraction over and stopped there → $\frac{9}{49}$

Answer: $8$

$11 \times 21 = 231$ is the largest multiple of $11$ that is not more than $239\mathclose{,}$ so the quotient is $21$ and the remainder is $239 - 231 = 8\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the quotient instead of the remainder → $21$
• Rounded the quotient up instead of down → $-3$

Answer: $\dfrac{10}{3}$

Write both fractions over the lowest common denominator, then combine the numerators: $\dfrac{9}{2} - \dfrac{7}{6} = \dfrac{27}{6} - \dfrac{7}{6} = \dfrac{20}{6} = \dfrac{10}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added (or subtracted) the numerators and the denominators separately → $- \frac{1}{2}$
• Wrote both over $6$ but did not change the numerators → $\frac{1}{3}$

Answer: $0.76$

Divide the numerator by the denominator: $19 \div 25 = 0.76\mathclose{.}$

Common mistakes, and the answer each one gives:

• Wrote the numerator and denominator as the digits after the point → $0.1925$

Answer: $3$

The bars mean the inside is $12$ away from zero in EITHER direction, so there are two equations: $6 x - 6 = 12$ gives $x = 3\mathclose{,}$ and $6 x - 6 = -12$ gives $x = -1\mathclose{.}$ The larger is $3\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $-1$
• Solved the positive branch, then negated the answer instead of solving the second branch → $-3$

Answer: $\dfrac{169}{264}$

Multiplication comes before addition, so work out $\dfrac{1}{6} \times \dfrac{1}{11} = \dfrac{1}{66}$ first, then add $\dfrac{5}{8}\mathclose{:}$ $\dfrac{169}{264}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{19}{264}$

Answer: $-40$

Division and multiplication have the same priority, so work from left to right: $-16 \div (-2) = 8\mathclose{,}$ then $8 \times (-5) = -40\mathclose{.}$ (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 → $40$
• Multiplied first, then divided → $- \frac{8}{5}$

Answer: $\dfrac{5}{2}$

Take the square root of the numerator and of the denominator: $\sqrt{\dfrac{25}{4}} = \dfrac{\sqrt{25}}{\sqrt{4}} = \dfrac{5}{2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Halved the fraction instead of taking its square root → $\frac{25}{8}$
• Took the square root of the numerator only → $\frac{5}{4}$

Answer: $\dfrac{31}{40}$

Write all three over the lowest common denominator, $40\mathclose{,}$ then combine the numerators: $\dfrac{4}{5} - \dfrac{1}{8} + \dfrac{1}{10} = \dfrac{32}{40} - \dfrac{5}{40} + \dfrac{4}{40} = \dfrac{31}{40}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added (or subtracted) the numerators and the denominators separately → $\frac{4}{7}$
• Wrote all three over the common denominator without changing the numerators → $\frac{1}{10}$
• Worked out the last two first, so the minus sign covered both → $\frac{23}{40}$

Answer: $\dfrac{68}{125}$

$0.544$ has 3 decimal places, so it is $\dfrac{544}{1000}\mathclose{;}$ dividing top and bottom by $8$ gives $\dfrac{68}{125}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $100$ as the denominator for 3 decimal places → $\frac{136}{25}$

Answer: $11$

$22 = 11 \times 2$ and $121 = 11 \times 11\mathclose{,}$ and $2$ and $11$ have no common factor left, so the greatest common divisor is $11\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the least common multiple instead → $242$

Answer: $\dfrac{77}{30}$

Write both fractions over the lowest common denominator, then combine the numerators: $\dfrac{7}{6} + \dfrac{7}{5} = \dfrac{35}{30} + \dfrac{42}{30} = \dfrac{77}{30}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added (or subtracted) the numerators and the denominators separately → $\frac{14}{11}$
• Wrote both over $30$ but did not change the numerators → $\frac{7}{15}$

Answer: $3$

The denominator was multiplied by $3$ $(8 \times 3 = 24)\mathclose{,}$ so the numerator must be too: $1 \times 3 = 3\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added to the top what was added to the bottom → $17$

Answer: $16$

Work out each inside first, then make it non-negative: $-5 - 12 = -17\mathclose{,}$ so the first bar gives $17\mathclose{;}$ $-9 + 8 = -1\mathclose{,}$ so the second gives $1\mathclose{.}$ Then $17 - 1 = 16\mathclose{.}$

Common mistakes, and the answer each one gives:

• Dropped the bars without making the insides positive → $-16$
• Applied the bars to each number on its own, as if $|a - b| = |a| - |b|$ → $-24$

Answer: $\dfrac{79}{120}$

Powers first, then multiplication, then subtraction: $\left(\dfrac{3}{4}\right)^{2} = \dfrac{9}{16}\mathclose{,}$ then $\dfrac{9}{16} \times \dfrac{6}{5} = \dfrac{27}{40}\mathclose{,}$ and $\dfrac{4}{3} - \dfrac{27}{40} = \dfrac{79}{120}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Worked strictly left to right, subtracting before multiplying → $\frac{37}{40}$
• Squared only the numerator → $- \frac{41}{30}$

Answer: $-\dfrac{27}{64}$

Raise the numerator and the denominator to the power $3\mathclose{:}$ $3^{3} = 27$ and $4^{3} = 64\mathclose{.}$ An odd power of a negative number is negative, so the answer is $-\dfrac{27}{64}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Dropped the minus sign: an odd power of a negative number is negative → $\frac{27}{64}$
• Multiplied by the exponent instead of raising to it → $- \frac{9}{4}$

Answer: $240$

Their greatest common divisor is $8\mathclose{,}$ and $\text{lcm} = \dfrac{40 \times 48}{8} = 240\mathclose{:}$ the smallest number both divide into.

Common mistakes, and the answer each one gives:

• Gave the greatest common divisor instead → $8$
• Multiplied the two numbers → $1920$

Answer: $\dfrac{64}{25}$

Dividing by a fraction is multiplying by it turned over: $\dfrac{8}{5} \div \dfrac{5}{8} = \dfrac{8}{5} \times \dfrac{8}{5} = \dfrac{64}{25}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied without turning the second fraction over → $1$
• Turned the FIRST fraction over instead of the second → $\frac{25}{64}$

Answer: $0.014$

Multiply as whole numbers, $7 \times 2 = 14\mathclose{,}$ then count the decimal places of BOTH factors — $1 + 2 = 3$ — so the answer is $0.014\mathclose{.}$

Common mistakes, and the answer each one gives:

• Counted the decimal places of only one factor → $0.14$