Unit 1: Multiplication
Topic B: Two- and Three-Digit Multipliers
When the multiplier is more than one digit, you use the same process and get partial products. You repeat the steps until you have multiplied by every digit, then add the partial products together.
Multiplying by Two-Digit Multipliers
Example A
[latex]24\times23=[/latex]
Part 1: Multiply by the ones digit in the multiplier.
Multiply 3 ones by 24 using the method you already know. The first partial product is 72.
[latex]\begin{array}{rr}&_1\hspace{0.5em}\\&24\\\times&23\\\hline&72\end{array}[/latex]
Part 2: Multiply by the tens digit in the multiplier. First, put a 0 to hold the ones place in your partial product. We are multiplying by a ten, so we hold the ones place.
Step 1: Multiply 2 tens by 4 ones = 8 tens
Write the 8 tens under the tens in your first partial product. It is very important to keep the columns straight – ones under one, tens under tens.
Step 2: Multiply 2 tens by 2 tens = 4 hundreds
Write the 4 hundreds in your partial product. The second partial product is 480.
[latex]\begin{array}{rr}&_1\hspace{0.5em}\\&24\\\times&23\\\hline&72\\&480\end{array}[/latex]
Part 3: Add the partial products together, being sure to add ones to ones, tens to tens, hundreds to hundreds. The sum is the final product.
Draw a line under the partial products. Add. Check your addition.
[latex]\begin{array}{rr}&_1\hspace{0.5em}\\&24\\\times&23\\\hline&72\\+&480\\\hline&552\end{array}[/latex]
Example B
[latex]36\times425 =[/latex]
Part 1: Multiply by the ones digit in the multiplier. 6 × 425 = 2550
[latex]\begin{array}{rr}&_1\hspace{0.15em}_3\hspace{0.5em}\\&425\\\times&36\\\hline&2550\end{array}[/latex]
Part 2: Multiply by the tens digit in the multiplier. First put a 0 to hold the ones place in the second partial product.
- Step 1: 3 tens × 5 tens = 15 tens = 1 hundred and 5 tens
Write the 5 tens in the second partial product and carry the 1 hundred. Now you can see why it is best to cross out the numbers you carry as soon as you have added them to the product. - Step 2: 3 tens × 2 tens = 6 hundreds
6 hundreds + 1 hundred (carried) = 7 hundreds. There is nothing to carry. - Step 3: 3 tens × 4 hundreds = 12 thousands
[latex]\begin{array}{rr}&_1\hspace{0.5em}\\&_1\hspace{0.15em}_3\hspace{0.5em}\\&425\\\times&36\\\hline&2550\\&12750\end{array}[/latex]
Part 3: Add the partial products together.
[latex]\begin{array}{rr}&_1\hspace{0.5em}\\&_1\hspace{0.15em}_3\hspace{0.5em}\\&425\\\times&36\\\hline&2550\\+&12750\\\hline&15300\end{array}[/latex]
tens × hundreds = thousands
Exercise 1
Multiply, being very careful to keep the columns straight when you write your partial products. Check your work using the answer key at the end of the exercise.
[latex]\begin{array}{rr}&84\\\times&12\\\hline&168\\+&840\\\hline&1008\end{array}[/latex]
- [latex]\begin{array}{rr}&73\\\times&12\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&50\\\times&42\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&62\\\times&31\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&61\\\times&42\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&91\\\times&53\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&92\\\times&31\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&91\\\times&49\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&72\\\times&48\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&53\\\times&30\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&41\\\times&53\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&42\\\times&94\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&80\\\times&86\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&31\\\times&79\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&54\\\times&40\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&61\\\times&48\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&60\\\times&31\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&55\\\times&73\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&84\\\times&56\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&53\\\times&38\\\hline\\\end{array}[/latex]
Answers to Exercise 1
- 876
- 2100
- 1922
- 2562
- 4823
- 2852
- 4459
- 3456
- 1590
- 2173
- 3948
- 6880
- 2449
- 2160
- 2928
- 1860
- 4015
- 4704
- 2014
When the multiplier has a zero in the ones place, use this shortcut.
Example C
[latex]\begin{array}{rr}&48\\\times&80\\\hline&3840\end{array}[/latex]
- Step 1: 0 ones × 48 = 0. Place one zero in the product and that will hold the ones place.
- Step 2: Multiply by the tens digit and write the product beside the zero.
Example D
[latex]\begin{array}{rr}&97\\\times&20\\\hline&1940\end{array}[/latex]
Exercise 2
Find the products. Use the shortcut for multipliers with a zero in them. Check your work using the answer key at the end of the exercise.
[latex]\begin{array}{rr}&76\\\times&70\\\hline&5320\end{array}[/latex]
- [latex]\begin{array}{rr}&52\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&91\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&83\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&49\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&61\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&16\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&36\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&398\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&432\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&863\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&907\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&503\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&452\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&943\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&248\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&6287\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&9025\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&8907\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&300\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&9075\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&3952\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&1528\\\times&70\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&7106\\\times&70\\\hline\\\end{array}[/latex]
Answers to Exercise 2
- 3640
- 6370
- 5810
- 3430
- 4270
- 1120
- 2520
- 27860
- 30240
- 60410
- 63490
- 35210
- 31640
- 66010
- 17360
- 440090
- 631750
- 623490
- 21000
- 635250
- 276640
- 106960
- 497420
Multiplying by Three-Digit Multipliers
To multiply by three digit multipliers, use the same method with one more part.
Example E
[latex]417\times368 =[/latex]
[latex]\begin{array}{rr}&417\\\times&368\\\hline&3336\\&25020\\+&125100\\\hline&153456\end{array}[/latex]
- Part 1: Multiply by the 8 ones.
- Part 2: Multiply the 6 tens; hold the ones place with 0.
- Part 3: Multiply by the 3 hundreds. Put 00 to hold the ones and tens places in the third partial product.
- Step 1: 3 hundreds × 7 ones = 21 hundreds = 2 thousands and 1 hundred. Write the 1 hundred and carry the 2 thousands.
- Step 2: 3 hundreds × 1 ten = 3 thousands. 3 thousands + 2 thousands (carried) = 5 thousands.
- Step 3: 3 hundreds × 4 hundreds = 12 ten thousands.
- Part 4: Add the partial products.
Exercise 3
Find the products. Check your work using the answer key at the end of the exercise.
- [latex]\begin{array}{rr}&416\\\times&213\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&375\\\times&291\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&361\\\times&475\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&275\\\times&863\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&984\\\times&469\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&489\\\times&578\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&498\\\times&123\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&267\\\times&854\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&613\\\times&368\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&725\\\times&547\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&269\\\times&912\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&752\\\times&697\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&983\\\times&357\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&835\\\times&148\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&386\\\times&296\\\hline\\\end{array}[/latex]
Answers to Exercise 3
- 88608
- 109125
- 171475
- 237325
- 461496
- 282642
- 61254
- 228018
- 225584
- 396575
- 245328
- 524144
- 350931
- 123580
- 114256
You know to hold the ones place with a zero if the multiplier has a zero in the ones place. Use the same skill if the multiplier has a zero in the tens place.
Example F
[latex]927\times405 =[/latex]
[latex]\begin{array}{rr}&927\\\times&405\\\hline&4635\\+&370800\\\hline&375435\end{array}[/latex]
- Part 1: Multiply by the 5 ones.
- Part 2: Multiply by the 0 tens.
Hold the ones place with a 0; 0 × 927 = 0; Place one zero in the tens place in the second partial product. - Part 3: Multiply by the 4 hundreds. The ones and tens places are already held by zeros. Start this partial product in the hundreds place on the same line.
- Part 4: Add the partial products.
Exercise 4
Find the products. Check your work using the answer key at the end of the exercise.
[latex]\begin{array}{rr}&_2\hspace{0.15em}_2\hspace{0.5em}\\&698\\\times&301\\\hline&698\\+&209400\\\hline&210098\end{array}[/latex]
- [latex]\begin{array}{rr}&923\\\times&403\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&830\\\times&108\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&482\\\times&206\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&432\\\times&205\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&625\\\times&405\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&275\\\times&306\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&765\\\times&506\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&1576\\\times&702\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&432\\\times&405\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&625\\\times&409\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&175\\\times&306\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&5874\\\times&309\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&7384\\\times&104\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&6538\\\times&603\\\hline\\\end{array}[/latex]
Answers to Exercise 4
- 371969
- 89640
- 99292
- 88560
- 255625
- 84150
- 387090
- 1106352
- 174960
- 255625
- 53550
- 1815066
- 767936
- 3942414
Multiplying by 10, 100, and 1000
Exercise 5
Do the following questions and see if you can find the pattern. Check your work using the answer key at the end of the exercise.
- [latex]\begin{array}{rr}&83\\\times&10\\\hline&830\end{array}[/latex]
- [latex]\begin{array}{rr}&46\\\times&10\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&97\\\times&10\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&123\\\times&10\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&70\\\times&10\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&129\\\times&10\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&1852\\\times&10\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&29871\\\times&10\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&45\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&26\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&432\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&679\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&2482\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&9037\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&46207\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&97512\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&23\\\times&1000\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&452\\\times&1000\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&207\\\times&1000\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&348\\\times&1000\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&2118\\\times&1000\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&2431\\\times&1000\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&23681\\\times&1000\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&48203\\\times&1000\\\hline\\\end{array}[/latex]
Answers to Exercise 5
- 830
- 460
- 970
- 1230
- 700
- 1290
- 18520
- 298710
- 4500
- 2600
- 43200
- 67900
- 248200
- 903700
- 4620700
- 9751200
- 23000
- 452000
- 207000
- 348000
- 2118000
- 2431000
- 23681000
- 48203000
And the pattern is …
When multiplying by 10, 100, 1000, 10000, etc., place as many zeros to the right of the number as there are zeros in the 10, 100, 1000, etc.
- To multiply by 10 put one zero after the number.
- To multiply by 100 put two zeros after the number.
- To multiply by 1000 put three zeros after the number.
Exercise 6
Find the products using the short method. Do not rewrite the questions. Check your work using the answer key at the end of the exercise.
- 12 × 10 = 120
- 10 × 3175 =
- 162 × 10 =
- 10 × 53821 =
- 10 × 123 =
- 27342 × 10 =
- 10 × 98 =
- 1134 × 10 =
- 15 × 100 =
- 100 × 278 =
- 9134 × 100 =
- 651 × 100 =
- 100 × 5169 =
- 100 × 24815 =
- 10 × 905 =
- 45683 × 10 =
- 1000 × 87 =
- 521 × 1000 =
- 1000 × 68935 =
- 1000 × 8902 =
- 1576 × 1000 =
- 31584 × 1000 =
- 1000 × 426 =
- 72 × 1000 =
Answers for Exercise 6
- 120
- 31750
- 1620
- 538210
- 1230
- 273420
- 980
- 11340
- 1500
- 27800
- 913400
- 65100
- 516900
- 2481500
- 9050
- 456830
- 87000
- 521000
- 68935000
- 8902000
- 1576000
- 31584000
- 426000
- 72000
Topic B: Self-Test
Mark /12 Aim 10/12
- Multiply these numbers.
- [latex]\begin{array}{rr}&47\\\times&39\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&58\\\times&93\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&48\\\times&100\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&982\\\times&1000\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&678\\\times&39\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&4579\\\times&86\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&8703\\\times&93\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&7390\\\times&85\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&8047\\\times&236\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&4238\\\times&197\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&8200\\\times&444\\\hline\\\end{array}[/latex]
- [latex]\begin{array}{rr}&7265\\\times&409\\\hline\\\end{array}[/latex]
Answers to Topic B Self-Test
- Multiply these numbers.
- 1833
- 5394
- 4800
- 982000
- 26442
- 393794
- 809379
- 628150
- 1899092
- 834886
- 3640800
- 3012285