KS2 Mathematics 2016 Paper 1: Arithmetic
Mark Scheme Legend
- 1m = 1 mark awarded for correct answer
- 2m = Up to 2 marks awarded (1m for correct method if answer incorrect)
- Method = Formal written method required for full marks on 2-mark questions
Table of Contents
- Question 1 (Addition)
- Question 2 (Addition)
- Question 3 (Division)
- Question 4 (Subtraction)
- Question 5 (Addition)
- Question 6 (Division)
- Question 7 (Addition)
- Question 8 (Subtraction)
- Question 9 (Division)
- Question 10 (Multiplication)
- Question 11 (Multiplication)
- Question 12 (Multiplication)
- Question 13 (Multiplication)
- Question 14 (Decimals)
- Question 15 (Division)
- Question 16 (Decimals)
- Question 17 (Decimals)
- Question 18 (Subtraction)
- Question 19 (Indices)
- Question 20 (Decimals)
- Question 21 (Decimals)
- Question 22 (Division)
- Question 23 (Long Mult)
- Question 24 (Fractions)
- Question 25 (Percentages)
- Question 26 (Decimals)
- Question 27 (Fractions)
- Question 28 (Long Div)
- Question 29 (Percentages)
- Question 30 (Long Mult)
- Question 31 (Fractions)
- Question 32 (Long Div)
- Question 33 (Fractions)
- Question 34 (Fractions)
- Question 35 (Fractions)
- Question 36 (BODMAS)
Question 1 (1 mark)
$$987 + 100 = \square$$
Worked Solution
Step 1: Understanding the Question
What do we need to do?
We are asked to add $100$ to $987$. Adding 100 changes the digit in the hundreds place by 1.
Step 2: Method
Strategy:
Look at the hundreds digit in $987$. It is $9$. If we add $1$ to the $9$ in the hundreds column, it becomes $10$ hundreds, which is $1000$.
✏ Working:
987 + 100 ----- 1087
Interpretation:
The unit digits (7) and tens digits (8) stay the same because we are adding 0 to them.
Final Answer:
1,087
✓ 1m
Question 2 (1 mark)
$$46 + 304 = \square$$
Worked Solution
Step 1: Understanding
We need to add two whole numbers. It is safest to use column addition to ensure we align the units correctly.
Step 2: Column Addition
Why we do this:
Align the units (6 and 4), tens (4 and 0), and hundreds (3). Remember to carry if the sum is 10 or more.
✏ Working:
304 + 46 ----- 350 1
$4 + 6 = 10$, so we write 0 and carry 1.
$0 + 4 + 1 \text{ (carry)} = 5$.
$3 + 0 = 3$.
Final Answer:
350
✓ 1m
Question 3 (1 mark)
$$326 \div 1 = \square$$
Worked Solution
Step 1: Understanding Division by 1
The Rule:
Any number divided by 1 is the number itself.
Step 2: Calculation
If you have 326 sweets and share them among 1 person, that person gets all 326 sweets.
$$326 \div 1 = 326$$
Final Answer:
326
✓ 1m
Question 4 (1 mark)
$$468 – 9 = \square$$
Worked Solution
Step 1: Understanding
We are subtracting a single digit number (9) from a three-digit number. We can do this mentally or use column subtraction.
Step 2: Method (Column Subtraction)
We cannot do $8 – 9$, so we must borrow from the tens column.
✏ Working:
45618 - 9 ------- 4 5 9
1. Borrow 10 from 60 (making it 50).
2. $18 – 9 = 9$.
3. Bring down the 5 and 4.
Final Answer:
459
✓ 1m
Question 5 (1 mark)
$$ \square = 936 + 285 $$
Worked Solution
Step 1: Setup
The empty box is on the left, but this just means we need to find the answer to $936 + 285$. We use column addition.
Step 2: Column Addition
✏ Working:
936 + 285 ----- 1221 11
1. $6 + 5 = 11$ (write 1, carry 1)
2. $3 + 8 + 1 = 12$ (write 2, carry 1)
3. $9 + 2 + 1 = 12$ (write 12)
Final Answer:
1,221
✓ 1m
Question 6 (1 mark)
$$95 \div 5 = \square$$
Worked Solution
Step 1: Method
Strategy:
We can use the “bus stop” method (short division) or split the number.
Let’s use the bus stop method.
✏ Working:
1 9 ----- 5| 945
1. How many 5s go into 9? 1 remainder 4.
2. Put the 4 in front of the 5 to make 45.
3. How many 5s go into 45? 9.
Final Answer:
19
✓ 1m
Question 7 (1 mark)
$$89,994 + 7,643 = \square$$
Worked Solution
Step 1: Alignment
Important:
Align the numbers to the right (units under units). Note that $89,994$ has 5 digits and $7,643$ has 4 digits.
Step 2: Column Addition
✏ Working:
89994 + 7643 ------- 97637 111
1. $4 + 3 = 7$
2. $9 + 4 = 13$ (write 3, carry 1)
3. $9 + 6 + 1 = 16$ (write 6, carry 1)
4. $9 + 7 + 1 = 17$ (write 7, carry 1)
5. $8 + 1 = 9$
Final Answer:
97,637
✓ 1m
Question 8 (1 mark)
$$ \square = 435 – 30 $$
Worked Solution
Step 1: Understanding
We are subtracting 3 tens (30) from the number 435. Look at the tens column.
Step 2: Calculation
Mental Method:
$435$ has $3$ tens. If we take away $3$ tens ($30$), we are left with $0$ tens.
$435 – 30 = 405$
Check with column method:
435 - 30 ---- 405
Final Answer:
405
✓ 1m
Question 9 (1 mark)
$$96 \div 4 = \square$$
Worked Solution
Step 1: Partitioning or Short Division
Method 1: Partitioning
Split 96 into numbers divisible by 4, like 80 and 16.
$$80 \div 4 = 20$$
$$16 \div 4 = 4$$
$$20 + 4 = 24$$
Step 2: Short Division
Method 2: Bus Stop
2 4 ----- 4| 916
1. 4 goes into 9 twice ($4 \times 2 = 8$), remainder 1.
2. Carry the 1 to make 16.
3. 4 goes into 16 four times.
Final Answer:
24
✓ 1m
Question 10 (1 mark)
$$879 \times 3 = \square$$
Worked Solution
Step 1: Short Multiplication
Multiply each digit of 879 by 3, starting from the units (right).
✏ Working:
8 7 9 x 3 ------- 2 6 3 7 2 2
1. $9 \times 3 = 27$ (write 7, carry 2)
2. $7 \times 3 = 21$. Add carry: $21 + 2 = 23$ (write 3, carry 2)
3. $8 \times 3 = 24$. Add carry: $24 + 2 = 26$ (write 26)
Final Answer:
2,637
✓ 1m
Question 11 (1 mark)
$$71 \times 8 = \square$$
Worked Solution
Step 1: Short Multiplication
We are multiplying a 2-digit number by a 1-digit number. We can do this mentally or write it down.
✏ Working:
7 1 x 8 ----- 5 6 8
1. $1 \times 8 = 8$
2. $7 \times 8 = 56$
Final Answer:
568
✓ 1m
Question 12 (1 mark)
$$50 \times 70 = \square$$
Worked Solution
Step 1: Using Related Facts
Strategy:
Multiply the non-zero digits first ($5 \times 7$), then account for the zeros.
$$5 \times 7 = 35$$
Step 2: Adjust for Place Value
Both numbers have been multiplied by 10 (5 became 50, 7 became 70).
So our answer must be multiplied by $10 \times 10 = 100$.
$$35 \times 100 = 3500$$
Alternative view: “Add the two zeros from the question to the end of 35”.
Final Answer:
3,500
✓ 1m
Question 13 (1 mark)
$$100 \times 412 = \square$$
Worked Solution
Step 1: Multiplying by 100
The Rule:
When multiplying a whole number by 100, the digits move two places to the left. This creates two zeros at the end.
Step 2: Apply Rule
Write down $412$.
Add two zeros to the end.
$$41200$$
Final Answer:
41,200
✓ 1m
Question 14 (1 mark)
$$3.005 + 6.12 = \square$$
Worked Solution
Step 1: Aligning Decimal Points
Crucial Step:
The decimal points must line up perfectly vertically. Fill empty spaces with zeros to avoid confusion.
$6.12$ is the same as $6.120$.
Step 2: Column Addition
✏ Working:
3.005 + 6.120 ------- 9.125
1. $5 + 0 = 5$
2. $0 + 2 = 2$
3. $0 + 1 = 1$
4. Drop decimal point
5. $3 + 6 = 9$
Final Answer:
9.125
✓ 1m
Question 15 (1 mark)
$$486 \div 3 = \square$$
Worked Solution
Step 1: Short Division
Use the bus stop method to divide 486 by 3.
✏ Working:
1 6 2 ------- 3| 418 6
1. 3 goes into 4 once, remainder 1.
2. Carry 1 to make 18. 3 goes into 18 six times ($3 \times 6 = 18$).
3. 3 goes into 6 twice ($3 \times 2 = 6$).
Final Answer:
162
✓ 1m
Question 16 (1 mark)
$$15.98 + 26.314 = \square$$
Worked Solution
Step 1: Alignment
Crucial Rule:
Align the decimal points vertically. Fill gaps with zeros to help with addition.
$15.98$ becomes $15.980$
Step 2: Column Addition
✏ Working:
15.980 + 26.314 -------- 42.294 1 1
1. $0 + 4 = 4$
2. $8 + 1 = 9$
3. $9 + 3 = 12$ (write 2, carry 1)
4. $5 + 6 + 1 = 12$ (write 2, carry 1)
5. $1 + 2 + 1 = 4$
Final Answer:
42.294
✓ 1m
Question 17 (1 mark)
$$125.48 – 72.3 = \square$$
Worked Solution
Step 1: Setup
Align the decimal points. Add a placeholder zero to $72.3$ to make it $72.30$.
Step 2: Column Subtraction
✏ Working:
01125.48 - 072.30 --------- 053.18
1. $8 – 0 = 8$
2. $4 – 3 = 1$
3. $5 – 2 = 3$
4. $2 – 7$ (can’t do), borrow from 1. $12 – 7 = 5$.
Final Answer:
53.18
✓ 1m
Question 18 (1 mark)
$$122,456 – 11,999 = \square$$
Worked Solution
Step 1: Strategy
Mental Trick: Subtracting $11,999$ is hard. Subtracting $12,000$ is easy.
$11,999$ is just $1$ less than $12,000$.
Step 2: Method (Option A: Mental Adjustment)
Subtract $12,000$:
$122,456 – 12,000 = 110,456$
We subtracted 1 too many, so add 1 back:
$110,456 + 1 = 110,457$
Step 3: Method (Option B: Column Subtraction)
✏ Working:
121213141516 - 1 1 9 9 9 -------------- 1 1 0 4 5 7
Lots of borrowing required here, making mistakes more likely!
Final Answer:
110,457
✓ 1m
Question 19 (1 mark)
$$3^2 + 10 = \square$$
Worked Solution
Step 1: Order of Operations (BODMAS)
We must do the index (power) first, before addition.
$3^2$ means “3 squared” or $3 \times 3$.
$$3 \times 3 = 9$$
Step 2: Addition
Now add the 10:
$$9 + 10 = 19$$
Final Answer:
19
✓ 1m
Question 20 (1 mark)
$$0.9 \div 10 = \square$$
Worked Solution
Step 1: Place Value Rule
The Rule:
When dividing by 10, digits move one place to the right. The number gets smaller.
Step 2: Apply Rule
Current: $0.9$ (9 tenths)
Move right: $0.09$ (9 hundredths)
Alternatively, move the decimal point one place to the left.
Final Answer:
0.09
✓ 1m
Question 21 (1 mark)
$$4 – 1.15 = \square$$
Worked Solution
Step 1: Alignment
Crucial Step:
Turn 4 into a decimal so it has the same number of decimal places as 1.15.
Write $4$ as $4.00$.
Step 2: Column Subtraction
✏ Working:
3 9
\cancel{4}.\cancel{1}010
- 1. 1 5
--------
2. 8 5
1. $0 – 5$ (can’t do). Borrow from tenths (which is 0). Borrow from units.
2. 4 becomes 3. First 0 becomes 10, then 9. Last 0 becomes 10.
3. $10 – 5 = 5$
4. $9 – 1 = 8$
5. $3 – 1 = 2$
Final Answer:
2.85
✓ 1m
Question 22 (1 mark)
$$1,320 \div 12 = \square$$
Worked Solution
Step 1: Recognising Multiples
We are dividing by 12. You might notice that 132 is a multiple of 12.
$$12 \times 11 = 132$$
Step 2: Calculation
Since $132 \div 12 = 11$:
$$1320 \div 12 = 110$$
Alternatively, use short division:
1 1 0
-------
12| 1 3 2 0
12 into 13 goes 1 remainder 1. 12 into 12 goes 1. 12 into 0 goes 0.
Final Answer:
110
✓ 1m
Question 23 (2 marks)
$$71 \times 46 = \square$$
Worked Solution
Step 1: Long Multiplication Setup
Method:
Multiply 71 by 6 (units).
Multiply 71 by 40 (tens) – remember the placeholder zero!
Add the two answers.
Step 2: Calculation
✏ Working:
7 1
x 4 6
-----
4 2 6 (71 x 6)
2 8 4 0 (71 x 40)
-------
3 2 6 6
1
Row 1: $1 \times 6 = 6$, $7 \times 6 = 42$.
Row 2: Place zero. $1 \times 4 = 4$, $7 \times 4 = 28$.
Sum: $6+0=6$, $2+4=6$, $4+8=12$ (carry 1), $2+1=3$.
Final Answer:
3,266
✓ 2m
Question 24 (1 mark)
$$\frac{4}{7} + \frac{5}{7} = \square$$
Worked Solution
Step 1: Adding Fractions
Rule:
Since the denominators (bottom numbers) are the same, we simply add the numerators (top numbers).
The denominator stays the same.
Step 2: Calculation
$$4 + 5 = 9$$
So the answer is:
$$\frac{9}{7}$$
Step 3: Convert to Mixed Number (Optional)
How many 7s go into 9? 1 remainder 2.
$$\frac{9}{7} = 1\frac{2}{7}$$
Both answers are acceptable.
Final Answer:
1 2/7 or 9/7
✓ 1m
Question 25 (1 mark)
$$20\% \text{ of } 1,800 = \square$$
Worked Solution
Step 1: Strategy
Method:
Find 10% first, then double it to find 20%.
Step 2: Find 10%
To find 10%, divide by 10.
$$1800 \div 10 = 180$$
Step 3: Find 20%
20% is double 10%.
$$180 \times 2 = 360$$
Final Answer:
360
✓ 1m
Question 26 (1 mark)
$$15 \times 6.1 = \square$$
Worked Solution
Step 1: Strategy
Method 1: Ignore Decimal Point
Treat it as $15 \times 61$, then put the decimal point back at the end.
Method 2: Partitioning
Calculate $15 \times 6$ and $15 \times 0.1$.
Step 2: Calculation (Method 1)
Calculate $15 \times 61$:
6 1 x 1 5 ----- 3 0 5 (5 x 61) 6 1 0 (10 x 61) ----- 9 1 5
The original question had 1 decimal place ($6.1$). So put 1 decimal place in the answer.
$$91.5$$
Step 3: Check (Method 2)
$15 \times 6 = 90$
$15 \times 0.1 = 1.5$
$90 + 1.5 = 91.5$
Final Answer:
91.5
✓ 1m
Question 27 (1 mark)
$$\frac{3}{10} – \frac{1}{20} = \square$$
Worked Solution
Step 1: Common Denominator
We cannot subtract fractions with different denominators (10 and 20).
We need to make the denominators the same. 20 is a multiple of 10.
Turn $\frac{3}{10}$ into twentieths.
Multiply top and bottom by 2:
$$\frac{3 \times 2}{10 \times 2} = \frac{6}{20}$$
Step 2: Subtraction
Now calculate:
$$\frac{6}{20} – \frac{1}{20} = \frac{5}{20}$$
Step 3: Simplify (Optional but recommended)
Divide top and bottom by 5:
$$\frac{5}{20} = \frac{1}{4}$$
Either answer is correct.
Final Answer:
1/4 or 5/20
✓ 1m
Question 28 (2 marks)
$$725 \div 29 = \square$$
Worked Solution
Step 1: List Multiples
For long division by 29, write down the first few multiples of 29:
$1 \times 29 = 29$
$2 \times 29 = 58$
$3 \times 29 = 87$ (Too big for 72)
Step 2: Long Division
✏ Working:
2 5
-----
29|7 2 5
-5 8
----
1 4 5
-1 4 5
------
0
1. 29 goes into 72 twice ($2 \times 29 = 58$).
2. Subtract: $72 – 58 = 14$.
3. Bring down the 5 to make 145.
4. How many 29s in 145? Try $29 \times 5$.
$20 \times 5 = 100$, $9 \times 5 = 45$. Total 145. Exact!
Final Answer:
25
✓ 2m
Question 29 (1 mark)
$$15\% \text{ of } 440 = \square$$
Worked Solution
Step 1: Breakdown
Method:
Find 10%. Find 5% (half of 10%). Add them together.
Step 2: Calculations
10% of 440: Divide by 10.
$$440 \div 10 = 44$$
5% of 440: Half of 10%.
$$44 \div 2 = 22$$
Total (15%):
$$44 + 22 = 66$$
Final Answer:
66
✓ 1m
Question 30 (2 marks)
$$6574 \times 31 = \square$$
Worked Solution
Step 1: Setup
Use long multiplication. Multiply by 1, then multiply by 30.
Step 2: Calculation
✏ Working:
6 5 7 4
x 3 1
---------
6 5 7 4 (x 1)
1 9 7 2 2 0 (x 30)
-----------
2 0 3 7 9 4
1 1
Row 1: Just write the number down (x1).
Row 2: Place zero. $4 \times 3 = 12$ (carry 1). $7 \times 3 = 21 + 1 = 22$ (carry 2). $5 \times 3 = 15 + 2 = 17$ (carry 1). $6 \times 3 = 18 + 1 = 19$.
Sum: Add the rows carefully.
Final Answer:
203,794
✓ 2m
Question 31 (1 mark)
$$1 \frac{4}{5} + \frac{3}{10} = \square$$
Worked Solution
Step 1: Common Denominator
Convert fifths into tenths so we can add them.
$$\frac{4}{5} = \frac{8}{10}$$
Step 2: Addition
$$1 \frac{8}{10} + \frac{3}{10}$$
Add the fractions:
$$\frac{8}{10} + \frac{3}{10} = \frac{11}{10}$$
Step 3: Convert and Combine
$$\frac{11}{10} = 1 \frac{1}{10}$$
Add this to the original whole number (1):
$$1 + 1 \frac{1}{10} = 2 \frac{1}{10}$$
Answer can also be $\frac{21}{10}$ or $2.1$.
Final Answer:
2 1/10
✓ 1m
Question 32 (2 marks)
$$1,118 \div 43 = \square$$
Worked Solution
Step 1: List Multiples of 43
Writing these down helps avoid mistakes.
$1 \times 43 = 43$
$2 \times 43 = 86$
$3 \times 43 = 129$
Step 2: Long Division
✏ Working:
2 6
-----
43|1 1 1 8
-8 6
----
2 5 8
-2 5 8
------
0
1. 43 into 111 goes twice ($2 \times 43 = 86$).
2. Subtract: $111 – 86 = 25$.
3. Bring down 8 to make 258.
4. How many 43s in 258? Let’s estimate: $40 \times 6 = 240$. Try 6.
$6 \times 3 = 18$, $6 \times 40 = 240$. Total 258. Exact!
Final Answer:
26
✓ 2m
Question 33 (1 mark)
$$\frac{3}{5} \div 3 = \square$$
Worked Solution
Step 1: Understanding Fraction Division
Concept:
If you have 3 pieces (fifths) and share them among 3 people, each person gets 1 piece (fifth).
Step 2: Method (KFC – Keep, Flip, Change)
$$\frac{3}{5} \div \frac{3}{1} = \frac{3}{5} \times \frac{1}{3}$$
Multiply across:
$$\frac{3 \times 1}{5 \times 3} = \frac{3}{15}$$
Step 3: Simplify
Divide top and bottom by 3:
$$\frac{1}{5}$$
Final Answer:
1/5
✓ 1m
Question 34 (1 mark)
$$\frac{2}{5} \times 140 = \square$$
Worked Solution
Step 1: Strategy
Divide by the bottom (denominator), multiply by the top (numerator).
Step 2: Divide by 5
$$140 \div 5$$
$100 \div 5 = 20$
$40 \div 5 = 8$
Total = 28
Step 3: Multiply by 2
$$28 \times 2 = 56$$
Final Answer:
56
✓ 1m
Question 35 (1 mark)
$$1 \frac{1}{4} – \frac{1}{3} = \square$$
Worked Solution
Step 1: Convert to Improper Fraction
It’s often easier to work with improper fractions.
$$1 \frac{1}{4} = \frac{5}{4}$$
Step 2: Common Denominator
The denominators are 4 and 3. The lowest common multiple is 12.
$$\frac{5}{4} = \frac{15}{12} \quad (\text{multiply by 3})$$
$$\frac{1}{3} = \frac{4}{12} \quad (\text{multiply by 4})$$
Step 3: Subtract
$$\frac{15}{12} – \frac{4}{12} = \frac{11}{12}$$
Final Answer:
11/12
✓ 1m
Question 36 (1 mark)
$$60 – 42 \div 6 = \square$$
Worked Solution
Step 1: Order of Operations (BODMAS)
Important: Division must be done before subtraction.
Step 2: Division
$$42 \div 6 = 7$$
Step 3: Subtraction
Now the equation is:
$$60 – 7 = 53$$
Final Answer:
53
✓ 1m
