Chapter 1: The Importance of Mental Math

Being proficient at mental arithmetic is invaluable. Whether you're calculating the total cost of your groceries before reaching the register or summing up your daily protein intake, enhancing your speed in mental calculations is always beneficial!

Consider a scenario where you’re evaluating a monthly subscription that costs £8. If you opt for an annual payment of £90, the service claims that this option offers a discount. To verify, you can determine the effective monthly price by dividing £90 by 12, which results in £7.50—a clear discount. While this example is simple, having a strategy becomes essential as numbers increase in complexity.

Chapter 2: A Step-by-Step Guide to Mental Division

In this section, I'll outline my method for performing mental division. Initially, this technique may seem applicable only when the dividend (the number being divided) is larger than the divisor. However, I have an effective workaround for that scenario, which I will explain later.

Step 1: Setting Up the Division

Let’s take two numbers, m and n, where n is smaller than m. For example, let’s use m = 1892 and n = 13. We can reframe the problem of dividing 1892 by 13 to ask, “What number multiplied by 13 equals 1892?” Our goal is to find that number.

To begin, we multiply n by 10 until it exceeds m. For our example:

We stop when we reach 13000, which is greater than 1892. For the next steps, we will only consider the numbers 13, 130, and 1300.

Step 2: Finding the Multiples

The essence of this step is to calculate multiples of 13 until we approach 1892. Even after 50 multiples of 13, we only reach 650. This is where calculating larger multiples like 10 * 13 and 100 * 13 becomes handy, as they serve as building blocks to reach our target number.

If you grasp the concept illustrated above, you might not need to delve deeper. However, here’s a detailed walkthrough:

From the numbers 13, 130, and 1300, we select the largest and multiply it until we surpass 1892. Starting from zero, we find:

When we reach 2600, which exceeds 1892, we stop. The last multiplication that resulted in a number less than 1892 was 1 * 1300. Thus, we keep track of:

• Current number: 1300
• Number breakdown: (100 * 13)

Next, we repeat this process with the second largest number, 130, adding multiples to our current total.

When we reach 1950, we note that it surpasses 1892, prompting us to record:

• Current number: 1820
• Number breakdown: (100 * 13) + (40 * 13) = (140 * 13)

Finally, we repeat the process for the last number, 13.

When we get to 1898, we find it exceeds 1892. Thus, our final tally is:

• Current number: 1885
• Number breakdown: (140 * 13) + (5 * 13) = (145 * 13)

At this point, we have exhausted our building blocks.

Step 3: Calculating the Remainder

The difference between 1885 and 1892 is 7, which gives us the remainder. Therefore, we can express our answer as:

Alternatively, you can state it as “145 remainder 7.”

I chose a more complex example to illustrate the method's effectiveness. With practice, you’ll become adept at recognizing known multiples of n, allowing you to streamline the process.

Handling Smaller Dividends

As I mentioned earlier, if the dividend is smaller than the divisor, simply multiply the dividend by 10 until it exceeds the divisor. After performing the division, divide your final result by 10 for each multiplication applied to the dividend.

While this technique may seem intricate at first, focusing on the visual explanation in Step 2 will greatly enhance your understanding. That part is crucial to mastering the method.

Challenge: Try These Calculations

Put your skills to the test with these mental challenges: 126/7, 252/11, and 999/8. There's a shortcut for the last one!