💡 What are we being asked to find?

We need to add three whole numbers together to find the total sum.

💡 Why we do this:

Using column addition ensures we add the digits in the correct place value (units with units, tens with tens, etc.) correctly.

🔍 What this tells us:

Starting from the right (units): \(5+1+9 = 15\) (write 5, carry 1).
Tens: \(5+0+4 + \text{carried } 1 = 10\) (write 0, carry 1).
Hundreds: \(1+5+6 + \text{carried } 1 = 13\) (write 3, carry 1).
Thousands: \(6 + \text{carried } 1 = 7\).

💡 What are we being asked to do?

We need to multiply a number by zero.

💡 Why we do this:

There is a special rule for multiplying by zero. “Zero times anything is zero.”

🔍 What this tells us:

If you have zero groups of 989, you have nothing.

💡 What are we being asked to find?

We need to find a number that, when added to 10, equals 302.

💡 Why we do this:

The inverse (opposite) of addition is subtraction. To find the missing part, we subtract the part we know (10) from the total (302).

🔍 What this tells us:

The missing number is 292. We can check by adding: \(10 + 292 = 302\).

💡 What are we being asked to do?

We need to divide 2,400 by 2. This is the same as finding half of 2,400.

💡 Why we do this:

When dividing numbers ending in zeros, we can divide the non-zero part first and then put the zeros back.

🔍 What this tells us:

Half of 2,400 is 1,200.

💡 What are we being asked to find?

We have a missing starting number. When we add 70 to it, we get 485.

💡 Why we do this:

To find the missing start number, we work backwards. The opposite of adding 70 is subtracting 70.

🔍 What this tells us:

The missing number is 415. Check: \(415 + 70 = 485\).

💡 What are we being asked to do?

We need to add two decimal numbers. The most important rule is to align the decimal points.

💡 Why we do this:

To avoid mistakes, we can add a ‘place holder’ zero to make the numbers the same length: 8.6 becomes 8.60.

🔍 What this tells us:

Hundredths: \(8 + 0 = 8\)
Tenths: \(4 + 6 = 10\) (write 0, carry 1)
Units: \(6 + 8 + 1 \text{ (carried)} = 15\)
The decimal point stays in the same straight line.

💡 What are we being asked to do?

We need to divide 240 by 8. The empty box is just asking for the answer.

💡 Why we do this:

We know that \(3 \times 8 = 24\). We can use this fact to solve \(240 \div 8\).

🔍 What this tells us:

The answer is 30.

💡 What are we being asked to do?

Divide 840 by 5. We can use the “Bus Stop” method (Short Division).

💡 Why we do this:

It breaks the large number into manageable chunks.

🔍 What this tells us:

1. How many 5s in 8? 1 remainder 3.
2. Carry the 3 to the 4 to make 34.
3. How many 5s in 34? 6 remainder 4.
4. Carry the 4 to the 0 to make 40.
5. How many 5s in 40? 8.

💡 What are we being asked to do?

Subtract 1,847 from 7,306.

💡 Why we do this:

When the top digit is smaller than the bottom digit, we need to “exchange” or “borrow” from the next column to the left.

🔍 What this tells us:

Units: \(6 – 7\) doesn’t work. Exchange from tens. Wait, tens is 0! Exchange from hundreds.
The 300 becomes 200, making the tens 10.
Now exchange from the 10 (making it 9) to make the units 16.
\(16 – 7 = 9\)
\(9 – 4 = 5\)
\(12 – 8 = 4\)
\(6 – 1 = 5\)

💡 What happens when we multiply by 10?

Multiplying a whole number by 10 shifts all digits one place to the left. We add a zero to the end.

💡 Why we do this:

It is efficient. \(1010\) becomes \(10100\).

💡 What do we know?

We need to divide 560 by 7. We can use our times tables to help us.

💡 Why we do this:

We know that \( 7 \times 8 = 56 \). Therefore, \( 56 \div 7 = 8 \).

🔍 What this tells us:

The answer is 80.

💡 Strategy:

We can multiply these numbers in any order. It is often easiest to multiply the single digits first, or leave the 10 for last.

💡 Why we do this:

Let’s do \( 6 \times 11 \) first, as that is a standard times table fact.

🔍 What this tells us:

The total product is 660.

💡 What is the best approach?

We can use short division (bus stop method) or recognize the relationship between 108 and 9.

🔍 What this tells us:

The answer is 120. (Check: \(12 \times 9 = 108\), so \(120 \times 9 = 1080\)).

💡 What are we subtracting?

We are taking 5 thousands away from 500 thousands. This is a mental calculation using place value.

💡 Why we do this:

Think of it as 500 (thousands) – 5 (thousands) = 495 (thousands).

🔍 What this tells us:

The zeros in the hundreds, tens, and units columns remain zeros. We only change the thousands and hundred-thousands columns.

💡 What method should we use?

We need to multiply a 3-digit number by a 1-digit number. Short multiplication (column method) is best here.

🔍 What this tells us:

1. \(6 \times 7 = 42\) (write 2, carry 4)
2. \(9 \times 7 = 63\). Add the 4 to get 67 (write 7, carry 6)
3. \(5 \times 7 = 35\). Add the 6 to get 41.

💡 What is the rule?

When we divide by 10, the digits shift one place to the right. The number gets 10 times smaller.

🔍 What this tells us:

The decimal point appears to move one step to the left.

💡 Strategy:

We are dividing by a 2-digit number (21). It is helpful to write down the first few multiples of 21: 21, 42, 63, 84…

🔍 What this tells us:

1. Divide: How many 21s in 67? \(3 \times 21 = 63\). So, 3 goes on top.
2. Subtract: \(67 – 63 = 4\).
3. Bring down: Bring down the 2 to make 42.
4. Divide: How many 21s in 42? \(2 \times 21 = 42\). So, 2 goes on top.
5. Subtract: \(42 – 42 = 0\). No remainder.

💡 Why can’t we just add them yet?

The fractions have different denominators (bottom numbers): 9 and 3. We must make them the same before adding. 9 is a multiple of 3, so we can use 9 as the common denominator.

🔍 What this tells us:

The answer is \(\frac{10}{9}\) (an improper fraction). We can also convert this to a mixed number: \(1 \frac{1}{9}\). Both answers are correct.

💡 Why do we need two rows?

We multiply by the units (3) in the first row, and by the tens (80) in the second row.

🔍 What this tells us:

Row 1: \(3 \times 7 = 21\) (1 carry 2). \(3 \times 0 = 0 + 2 = 2\). \(3 \times 6 = 18\). Result: 1821.
Row 2: Place a zero placeholder. \(8 \times 7 = 56\) (6 carry 5). \(8 \times 0 = 0 + 5 = 5\). \(8 \times 6 = 48\). Result: 48560.
Sum: Add the two rows to get 50,381.

💡 What happens when we multiply by 1,000?

The number gets 1,000 times bigger. All digits move 3 places to the left.

🔍 What this tells us:

The decimal point moves 3 jumps to the right. We need to add a placeholder zero to fill the gap.

💡 What do we notice?

Both fractions have the same denominator (3). This makes addition easy. We can add the whole numbers and the fractions separately.

🔍 What this tells us:

The total is exactly 3.

💡 How do we find a fraction of an amount?

We divide the amount by the denominator (bottom number) and multiply by the numerator (top number).

“Divide by the bottom, times by the top.”

🔍 What this tells us:

If we split 30 into 10 parts, each part is 3. We want 7 of those parts, which is 21.

💡 The trick:

The number 8 is a whole number. To subtract a decimal from it, we must write 8 as 8.000 so we have placeholders to subtract from.

🔍 What this tells us:

We have to exchange across all the zeros.
\(10 – 3 = 7\)
\(9 – 2 = 7\)
\(9 – 1 = 8\)
\(7 – 5 = 2\)

💡 What does this mean?

We have one eighth, and we need to cut it in half (divide by 2). If you cut a small slice of cake in half, the piece gets smaller.

💡 The Rule:

Dividing a fraction by an integer is the same as multiplying the denominator by that integer.

🔍 What this tells us:

Half of an eighth is a sixteenth.

💡 Why is this harder?

The denominators (2 and 3) are different. We need a number that both 2 and 3 go into.

The lowest common multiple of 2 and 3 is 6.

💡 Important Setup:

Just like Question 23, we are subtracting a decimal from a whole number. We must add placeholders to the whole number 26.

🔍 What this tells us:

We exchange from the 6 (making it 5), passing the 1 across the zeros.
\(10 – 2 = 8\)
\(9 – 1 = 8\)
\(9 – 0 = 9\)
\(5 – 2 = 3\)
\(2 – 0 = 2\)

💡 Strategy:

We can’t easily find 15% in one go. But we can find 10% and 5% easily.

\( 15\% = 10\% + 5\% \)

🔍 What this tells us:

The total value is 480.

💡 Strategy:

To find 2%, first find 1%. To find 1%, we divide the total by 100.

🔍 What this tells us:

2% of 3,000 is 60.

💡 Strategy:

We are dividing by 73. This is tricky! It helps to write a “coin list” or estimate.
\(73 \times 1 = 73\)
\(73 \times 2 = 146\)
\(73 \times 10 = 730\)

🔍 What this tells us:

1. 73 into 30 doesn’t go.
2. 73 into 306? Try 4. \(70 \times 4 = 280\), so close.
\(73 \times 4 = 292\). Remainder is \(306 – 292 = 14\).
3. Bring down the 6 to make 146.
4. 73 into 146? Double 73 is 146. So, it goes exactly 2 times.

💡 Strategy:

If we find 10%, we can multiply it by 8 to get 80%.

Alternative Method: Find 20% (double 10% = 23) and subtract it from the total (100% – 20% = 80%).
\(115 – 23 = 92\).

💡 Why is this hard?

The denominators are 7 and 9. We need to find a number that is in both the 7 and 9 times tables.

\(7 \times 9 = 63\). We will use 63.

💡 Strategy:

Subtracting from a mixed number is easier if we turn it into an improper (top-heavy) fraction first.

💡 Strategy:

Multiply 4078 by 7, then multiply 4078 by 60, then add.

🔍 What this tells us:

Row 1: \(7 \times 8 = 56\). \(7 \times 7 = 49 (+5) = 54\). \(7 \times 0 = 0 (+5) = 5\). \(7 \times 4 = 28\).
Row 2: Zero placeholder. \(6 \times 8 = 48\). \(6 \times 7 = 42 (+4) = 46\). \(6 \times 0 = 0 (+4) = 4\). \(6 \times 4 = 24\).
Total: \(28,546 + 244,680 = 273,226\).

💡 Method:

We can subtract the whole number first, then the fraction.

🔍 What this tells us:

The answer is \(7\frac{3}{4}\).

💡 The Golden Rule:

You must Divide before you Add. (Division comes before Addition in BIDMAS).

⚠️ Common Mistake:

If you add first (\(6+4=10\)) then divide (\(10 \div 2 = 5\)), the answer is wrong.

💡 Strategy:

This is the same as finding \(\frac{4}{5}\) of 400.

Divide by the denominator (5), then multiply by the numerator (4).

🔍 What this tells us:

The answer is 320.