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Number Toolkit

Number

Column multiplication

  • Multiply the top number by each digit of the bottom number in turn, working right to left
  • Write a zero as a place holder for each column moved leftwards through the bottom number
  • Add the partial products to reach the final answer

The grid method

  • The grid method splits each number into its place-value parts, so no carrying is needed until the final addition
  • For 246 × 34, split 246 into 200, 40 and 6, and 34 into 30 and 4
×200406
306 0001 200180
480016024
  • Adding every cell gives 6 000 + 1 200 + 180 + 800 + 160 + 24 = 8 364

The lattice method

  • Every product stays a single-digit pair in this layout, which is what makes it reliable for longer numbers
  • Each cell is split by a diagonal, with the tens digit above the diagonal and the ones digit below
  • The digits are then summed along each diagonal, carrying leftwards where a diagonal totals ten or more
Lattice grid for 2879 multiplied by 36: the digits 2, 8, 7, 9 label the columns and 3, 6 label the rows, each cell is split by a diagonal with the tens digit above and the ones digit below, and the diagonals are summed down the left edge and along the bottom to read off 103 644
Source: Multiplication by Save My Exams

Short division

  • Short division divides by a single digit, carrying any remainder into the next digit of the number
  • The remainder at each step is written as a small digit in front of the next digit
Short division of 568 by 8: the divisor 8 sits outside the bracket and 568 inside, a small raised 5 records the remainder carried from the first step onto the next digit, and the quotient 071 is written above the line
Source: Multiplication by Save My Exams
  • Only one written method for each operation needs to be known, provided it is used accurately

Shortcuts worth spotting

  • Multiplying or dividing by a power of ten shifts every digit along the place-value columns, one column for each zero
    • 380 ÷ 10 = 38, 45 ÷ 100 = 0.45 and 28 ÷ 1000 = 0.028
    • 3.7 × 100 = 370
  • Writing the leading zeros in first makes the shift easier to see, so 0028 ÷ 1000 becomes 0.028
  • A division by a larger number can be simplified first, exactly as a fraction is cancelled
    • 1008 ÷ 28 is 1008/28, which halves to 504/14 and again to 252/7, and short division finishes it as 36
  • Halving repeatedly divides by a power of 2, so 568 ÷ 8 is 284, then 142, then 71
Worked example

A long multiplication and a short division

Without a calculator, work out 384 × 27 and 4 718 ÷ 7.

Solution:

  • For the multiplication, split 27 into 20 and 7 and multiply separately
  • 384 × 20 = 7 680
  • 384 × 7 = 2 688
  • Add the partial products: 7 680 + 2 688 = 10 368
  • Estimate as a check: 400 × 27 = 10 800, so 10 368 is sensible
  • For the division, work left to right carrying each remainder
  • 7 into 4 does not go, so carry the 4 onto the next digit to make 47
  • 7 into 47 goes 6 times with 5 left over, so write 6 and carry the 5 onto the 1 to make 51
  • 7 into 51 goes 7 times with 2 left over, so write 7 and carry the 2 onto the 8 to make 28
  • 7 into 28 goes 4 times exactly
  • The answer is 674, and 7 × 674 = 4 718 confirms it