Multiply 23 by 47 with shifted partial products.
Step 1 — Set up
Set up the multiplication board.
x x x 2 3 × x x 4 7 x x □ □ □ x x □ □ □ x □ □ □ □ \begin{array}{rrrrr}\phantom{x}&\phantom{x}&\phantom{x}&2&3\\\times&\phantom{x}&\phantom{x}&4&7\\\hline\\\phantom{x}&\phantom{x}&\square&\square&\square\\\phantom{x}&\phantom{x}&\square&\square&\square\\\hline\\\phantom{x}&\square&\square&\square&\square\end{array} x × x x x x x x x □ x x □ □ □ 2 4 □ □ □ 3 7 □ □ □ Set up 23 x 47 aligned partials -- -- 2 3 x -- -- 4 7 -- -- -- -- -- -- -- -- -- -- -- -- ones row, shifted tens row, then sum ones row: x 7 7 x 3 = 21, write 1 and carry 2 7 x 2 tens + carry 2 = 16 tens; row 161 tens row: x 4 tens 4 x 3 = 12, write 2 and carry 1 4 x 2 tens + carry 1 = 9 tens raw 92, shift one place to 920 add aligned partials -- + -- = --
Step 2 — Ones partial
Ones digit 7: 3 x 7 = 21, write 1, carry 2; partial 161.
x x x 2 x x x x 2 3 × x x 4 7 x x 1 6 1 x x □ □ □ x □ □ □ □ \begin{array}{rrrrr}\phantom{x}&\phantom{x}&\phantom{x}&\hl{2}&\phantom{x}\\\phantom{x}&\phantom{x}&\phantom{x}&2&3\\\times&\phantom{x}&\phantom{x}&4&7\\\hline\\\phantom{x}&\phantom{x}&\hl{1}&\hl{6}&\hl{1}\\\phantom{x}&\phantom{x}&\square&\square&\square\\\hline\\\phantom{x}&\square&\square&\square&\square\end{array} x x × x x x x x x x x □ x x x 1 □ □ 2 2 4 6 □ □ x 3 7 1 □ □ 7 x 3 = 21: write 1, carry 2 aligned partials 2 carry -- -- 2 3 x -- -- 4 7 -- 1 6 1 -- -- -- -- -- -- -- -- ones row, shifted tens row, then sum ones row: x 7 7 x 3 = 21, write 1 and carry 2 7 x 2 tens + carry 2 = 16 tens; row 161 tens row: x 4 tens 4 x 3 = 12, write 2 and carry 1 4 x 2 tens + carry 1 = 9 tens raw 92, shift one place to 920 add aligned partials -- + -- = --
Step 3 — Tens partial
Tens digit 4: 3 x 4 = 12, write 2, carry 1; shift partial to 920.
x x x 1 x x x x 2 3 × x x 4 7 x x 1 6 1 x x 9 2 0 x □ □ □ □ \begin{array}{rrrrr}\phantom{x}&\phantom{x}&\phantom{x}&\hl{1}&\phantom{x}\\\phantom{x}&\phantom{x}&\phantom{x}&2&3\\\times&\phantom{x}&\phantom{x}&4&7\\\hline\\\phantom{x}&\phantom{x}&1&6&1\\\phantom{x}&\phantom{x}&\hl{9}&\hl{2}&\hl{0}\\\hline\\\phantom{x}&\square&\square&\square&\square\end{array} x x × x x x x x x x x □ x x x 1 9 □ 1 2 4 6 2 □ x 3 7 1 0 □ 4 tens x 23 gives 920 aligned partials 1 carry -- -- 2 3 x -- -- 4 7 -- 1 6 1 -- 9 2 0 -- -- -- -- ones row, shifted tens row, then sum ones row: x 7 7 x 3 = 21, write 1 and carry 2 7 x 2 tens + carry 2 = 16 tens; row 161 tens row: x 4 tens 4 x 3 = 12, write 2 and carry 1 4 x 2 tens + carry 1 = 9 tens raw 92, shift one place to 920 add aligned partials -- + -- = --
Step 4 — Add partials
Add partials: 161 + 920 = 1081.
x x x 2 3 × x x 4 7 x x 1 6 1 x x 9 2 0 x 1 0 8 1 \begin{array}{rrrrr}\phantom{x}&\phantom{x}&\phantom{x}&2&3\\\times&\phantom{x}&\phantom{x}&4&7\\\hline\\\phantom{x}&\phantom{x}&1&6&1\\\phantom{x}&\phantom{x}&9&2&0\\\hline\\\phantom{x}&\hl{1}&\hl{0}&\hl{8}&\hl{1}\end{array} x × x x x x x x x 1 x x 1 9 0 2 4 6 2 8 3 7 1 0 1 161 + 920 = 1081 aligned partials -- -- 2 3 x -- -- 4 7 -- 1 6 1 -- 9 2 0 1 0 8 1 ones row, shifted tens row, then sum ones row: x 7 7 x 3 = 21, write 1 and carry 2 7 x 2 tens + carry 2 = 16 tens; row 161 tens row: x 4 tens 4 x 3 = 12, write 2 and carry 1 4 x 2 tens + carry 1 = 9 tens raw 92, shift one place to 920 add aligned partials 161 + 920 = 1081
Step 5 — Result
Read the complete product: 1081.
x x x 2 3 × x x 4 7 x x 1 6 1 x x 9 2 0 x 1 0 8 1 \begin{array}{rrrrr}\phantom{x}&\phantom{x}&\phantom{x}&2&3\\\times&\phantom{x}&\phantom{x}&4&7\\\hline\\\phantom{x}&\phantom{x}&1&6&1\\\phantom{x}&\phantom{x}&9&2&0\\\hline\\\phantom{x}&1&0&8&1\end{array} x × x x x x x x x 1 x x 1 9 0 2 4 6 2 8 3 7 1 0 1 23 x 47 = 1081 aligned partials -- -- 2 3 x -- -- 4 7 -- 1 6 1 -- 9 2 0 1 0 8 1 ones row, shifted tens row, then sum ones row: x 7 7 x 3 = 21, write 1 and carry 2 7 x 2 tens + carry 2 = 16 tens; row 161 tens row: x 4 tens 4 x 3 = 12, write 2 and carry 1 4 x 2 tens + carry 1 = 9 tens raw 92, shift one place to 920 add aligned partials 161 + 920 = 1081