Lesson 67: Multiplying by Multiples of 10

New Concept

To multiply a whole number or a decimal number by a multiple of 10, we may write the multiple of 10 so that the zero 'hangs out' to the right.

What’s next

Next, you’ll apply this 'hang out' method to multiply whole numbers, decimals, and solve word problems.

To multiply a number by a multiple of 10, like 20, you can first break the multiple into its factors. For example, since 20 is equal to $2 \times 10$, you can solve $34 \times 20$ by calculating $34 \times 2 \times 10$. This turns one big multiplication problem into two smaller, more manageable steps.

Example 1: Solve $25 \times 30$ by factoring. First, rewrite 30 as $3 \times 10$. The problem becomes $25 \times 3 \times 10$. Then, calculate $25 \times 3 = 75$. Finally, multiply by 10 to get $75 \times 10 = 750$. Example 2: To solve $12 \times 30$, we can express it as $12 \times 3 \times 10$. First, $12 \times 3 = 36$. Then, $36 \times 10 = 360$.

Think of this as a strategic detour! Instead of tackling a big number like 40 all at once, you can multiply by 4 first, which is much easier, and then multiply that result by 10. It’s like breaking a long journey into two short, easy trips, ensuring you arrive at the correct answer without getting lost in large numbers.

When multiplying by a multiple of 10 vertically, you can write the problem so the zero 'hangs out' to the right. First, bring the hanging zero straight down into the answer. Then, simply multiply the remaining numbers as you normally would. This is a quick and organized shortcut for these types of problems.

Example 1: To calculate $34 \times 20$, write the 20 below the 34 with the 0 hanging out. Drop the 0 into the answer. Then, multiply $34 \times 2 = 68$. The final answer is $680$. Example 2: For $45 \times 50$, let the 0 hang out and drop it down. Then, calculate $45 \times 5 = 225$. Your result is $2250$. Example 3: Solving $409 \times 70$, you drop the 0 and then multiply $409 \times 7 = 2863$. The final answer is $28630$.

Give that zero a break and let it hang out! This cool trick cleans up your workspace by letting you drop the zero straight down into the answer line. After that, you can forget about it and focus on the simpler multiplication problem that’s left over. It’s a fantastic way to keep your calculations neat and avoid simple mistakes.

To multiply a decimal number by a multiple of 10, use the 'hanging zero' method. After you get a product, you must place the decimal point correctly. Count the number of digits after the decimal point in the original decimal number, and make sure your final answer has the same number of decimal places.

Example 1: To find the total cost of 20 items at 1.43 dollars each, multiply $1.43 \times 20$. Let the zero hang out and drop down. Then, $143 \times 2 = 286$. Since 1.43 has two decimal places, the answer is 28.60 dollars. Example 2: For $1.64 \times 30$, drop the 0 and multiply $164 \times 3 = 492$. Since 1.64 has two decimal places, the answer is 49.20 dollars.

This is the same 'hanging zero' party, but with a special guest: the decimal point! Do your multiplication just as you learned, then stop and count the decimal places in the number you started with. Your final answer must have the exact same number of places after the decimal. It ensures your dollars and cents are always on point!