Multiplying Digits with 5 and 10 Up to 999

Multiplying by 5 and multiplying by 10 are two of the easiest facts in all of multiplication, because both numbers connect directly to our base ten place value system. Once you see the pattern, you can multiply any whole number up to 999 by 5 or by 10 almost instantly, without needing to add repeated groups one at a time.

This topic builds on ideas from times tables and on the groups-based thinking from repeated addition as multiplication. If you have not yet looked at the simplest facts, it also helps to review multiplying digits with 0 and 1 up to 999 first, since those facts follow a similar "look for the pattern" approach.

Every number in our place value system is made of ones, tens, and hundreds. When you multiply a whole number by 10, each digit moves one place value to the left, and a 0 fills the empty ones place. This is why multiplying by 10 always looks like "add a zero to the end."

For example, \( 8 \times 10 = 80 \), \( 34 \times 10 = 340 \), and \( 96 \times 10 = 960 \). Even with larger numbers this still works, as long as the result stays under 1000: \( 99 \times 10 = 990 \).

Think of it this way: multiplying by 10 does not create new digits out of nowhere. It simply moves the digits you already have one place value up, from ones to tens, or from tens to hundreds.

Multiplying by 5 does not have the same neat "add a zero" trick, but it has an even more useful shortcut once you notice it: multiplying by 5 always gives half of multiplying the same number by 10.

In symbols, \( n \times 5 = \dfrac{n \times 10}{2} \). So to find \( 34 \times 5 \), first find \( 34 \times 10 = 340 \), then take half of that: \( 340 \div 2 = 170 \). That means \( 34 \times 5 = 170 \).

You can also think of multiplying by 5 as skip counting by fives: 5, 10, 15, 20, 25, and so on. The nth number in that skip counting sequence is exactly \( n \times 5 \). For a group amount like 5 stickers per page, counting pages by fives gets you the total quickly without a repeated addition table.

Spotting the pattern in a chart

Lining up a few facts side by side makes both patterns easy to see at once.

Example 1: Find \( 78 \times 10 \). Shift every digit one place value left and place a 0 in the ones place: \( 78 \times 10 = 780 \).

Example 2: Find \( 62 \times 5 \). First multiply by 10: \( 62 \times 10 = 620 \). Then take half: \( 620 \div 2 = 310 \). So \( 62 \times 5 = 310 \).

Example 3: A bakery packs muffins into boxes of 5. If there are 84 boxes, how many muffins are there? This is \( 84 \times 5 \). Since \( 84 \times 10 = 840 \), half of that is \( 420 \). There are 420 muffins in total.

Example 4: A farmer plants seeds in rows of 10. With 56 rows, the total number of seeds is \( 56 \times 10 = 560 \).

Once you have an answer, a quick check is to look at the last digit. Any whole number multiplied by 10 must end in 0. Any whole number multiplied by 5 must end in either 0 or 5, since 5 times an even number ends in 0, and 5 times an odd number ends in 5. If your answer breaks either rule, it is worth going back and checking your multiplication.

As you get comfortable with these two patterns, it is a natural next step to try multiplying using mental strategies, where the same kind of place value thinking is used to multiply by other numbers without a written algorithm.

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