Operation order, inverses and reciprocals

(a) 8

1. The order of operations puts multiplication before subtraction.
2. 2 $\times$ 5 = 10, so 18 $-$ 10 = 8.

• B1 Correct answer: 8

(a) $8$

1. Multiply before subtracting: $3 \times 4 = 12$, then $20 - 12 = 8$.

• B1 The correct answer, $8$.

(b) $12$

1. Brackets first, then the power.
   $$(5 - 8)^2 = (-3)^2 = 9$$
2. Divide before adding.
   $$9 + 6 \div 2 = 9 + 3 = 12$$

• M1 Finding $(-3)^2 = 9$ or $6 \div 2 = 3$.
• A1 The correct answer, $12$.

(c) $\frac{1}{4}$

1. The reciprocal of a number is $1$ divided by it: $\frac{1}{4}$, or $0.25$.

• B1 $\frac{1}{4}$ or $0.25$.

(a) $17$; Maya subtracted before dividing.

1. Division comes before addition and subtraction: $6 \div 3 = 2$, so $18 - 2 + 1 = 17$.
2. Maya worked from left to right: $18 - 6 = 12$, $12 \div 3 = 4$, $4 + 1 = 5$. She subtracted before dividing.

• B1 The correct value, $17$.
• C1 Saying Maya did the subtraction $18 - 6$ before the division $6 \div 3$.

(b) $(18 - 6) \div 3 + 1$

1. $(18 - 6) \div 3 + 1 = 12 \div 3 + 1 = 4 + 1 = 5$.

• B1 Choosing $(18 - 6) \div 3 + 1$.

(c) $24$

1. Dividing by a number is the same as multiplying by its reciprocal.
2. $$3 \div 0.125 = 3 \times 8 = 24$$

• M1 Writing $3 \times 8$.
• A1 The correct answer, $24$.

(a) $53$

1. $$11-7$$
2. $$3\times 16$$
3. Work inside the brackets first: 11 $-$ 7 = 4.
4. Calculate the power, then multiplication: $4^{2}$ = 16 and 3 $\times$ 16 = 48.
5. Finally add: 5 + 48 = 53.

• M1 Establishing $11-7$ or an equivalent valid method.
• M1 Establishing $3\times 16$ or an equivalent valid method.
• A1 Correct answer: $53$

(a) $4$

1. Find A’s output, applying its operations in order.
   $$(7+2)\times 3$$
2. Subtract B’s output.
   $$27-(7\times 3+2)$$
3. Therefore $4$.

• P1 Find A’s output, applying its operations in order.
• P1 Subtract B’s output.
• A1 Correct answer: $4$

(b) $\frac{1}{3}$

1. Therefore $\frac{1}{3}$.

• B1 Correct answer: $\frac{1}{3}$

(a) $\frac{121}{4}$

1. Evaluate the bracket before squaring.
   $$8+3$$
2. Square the bracket value and divide by 4.
   $$11^{2}/4$$
3. Therefore $\frac{121}{4}$.

• M1 Evaluate the bracket before squaring.
• M1 Square the bracket value and divide by 4.
• A1 Correct answer: $\frac{121}{4}$

(b) $\frac{4}{121}$

1. Invert the exact non-zero result.
   $$4/121$$
2. Therefore $\frac{4}{121}$.

• M1 Invert the exact non-zero result.
• A1 Correct answer: $\frac{4}{121}$

(a) $18$

1. Work out the bracket, then divide and multiply from left to right.
   $$36/6\times 3$$
2. Therefore $18$.

• M1 Work out the bracket, then divide and multiply from left to right.
• A1 Correct answer: $18$

(b) $\frac{5}{2}$

1. Therefore $\frac{5}{2}$.

• B1 Correct answer: $\frac{5}{2}$

(c) $17.5$

1. Dividing by 0.4 is the same as multiplying by its reciprocal.
   $$7\times 2.5$$
2. Therefore $17.5$.

• M1 Dividing by 0.4 is the same as multiplying by its reciprocal.
• A1 Correct answer: $17.5$

(a) $-3$ or $\frac{23}{3}$

1. Undo the operations in reverse order. The last step was squaring, and both $4^2$ and $(-4)^2$ are $16$.
2. Multiply each by 4.
   $$4 \times 4 = 16, \qquad -4 \times 4 = -16$$
3. Add 7.
   $$16 + 7 = 23, \qquad -16 + 7 = -9$$
4. Divide by 3.
   $$23 \div 3 = \frac{23}{3}, \qquad -9 \div 3 = -3$$

• P1 Undoing the square to get both $4$ and $-4$.
• P1 Undoing the division and the subtraction in reverse order, reaching $23$ or $-9$.
• P1 Dividing by 3 at the end, for either value.
• A1 Both $-3$ and $\frac{23}{3}$ (or $7\frac{2}{3}$).