Tips for Flash Mental Math with Addition and Subtraction: Fast, Accurate Mixed Problems

Three reasons mixed problems are harder than addition

• A switching decision is added: each time you see a number you have to decide “add or subtract”. At speeds like 0.5 seconds per number, there’s almost no time for that decision.
• Missing the sign: if you look only at the number and miss the “−”, you add that number. The answer then ends up off from the correct one by twice that number.
• Borrowing comes in: on top of carrying in addition, you now have borrowing from the place above. On the abacus, that means more moves using the complement of 10.

None of these mean your addition isn’t good enough; you just aren’t used to the moves specific to mixed problems. That’s why, rather than switching to mixed problems at the same conditions as addition, the quickest route is to ease the conditions first and get used to it.

When to move on to mixed problems, and the conditions to start with

A good sign is when you can get 8 or more out of 10 right on addition at the same number of digits, at around 0.6 seconds per number. When you switch to mixed problems, keep the count the same and start with about 1.5 times the seconds per number.

In this site’s flash mental math tool, setting “Operation” to “Addition & subtraction” gives mixed problems. The first number is always added, and it never gives a subtraction that would make the running total negative. With a count of 3 or more, at least one number is subtracted. There are never problems where you’d run out of beads on the abacus, so you can practice with confidence.

How to subtract using complements

On the abacus, when you can’t move a number directly to add or subtract it, you use its “complement”: the number that adds up with it to make 5 or 10. Thinking the same way in mental math lets you handle borrowing in an instant.

1. 1

   Subtract with the complement of 10 (borrowing)

   When you can’t subtract from the ones place, as in 13 − 7, take 10 from the tens place and add 7’s complement, 3, to the ones place. 13 → reduce the tens by 1 to get 3 → 3 + 3 = 6. Remember it as “subtracting 7 = take away 10, add 3”.

2. 2

   Subtract with the complement of 5 (the abacus’s 5-bead)

   On the abacus, when you can’t subtract with the 1-beads alone, as in 6 − 2, clear the 5-bead (−5) and add 2’s complement of 5, which is 3, with the 1-beads (+3). Make the same movement on the abacus in your head.

3. 3

   Combine the two

   Sometimes, as in 12 − 6, you borrow from the tens place and also need a 5-bead move. Once you know these as bead movements, your hands (and head) will move the moment you see the number.

Tips for not missing the minus sign

• Look at the sign before the number: build the habit of first checking whether there’s a “−” to the left of the number. The tool shows numbers to subtract in red with a “−”, so you can also tell by color.
• Use the sound as a cue: the tool’s beep is a little lower for numbers to subtract. You know “subtraction next” by ear before looking, which makes it easier to switch in time.
• Keep the total after subtracting in mind: after a subtraction, the total gets smaller. With a sense that “it should have gone down”, you’ll notice something feels off if you added by mistake.

In listening arithmetic, a warning comes before each number to subtract. That makes it easier to prepare for the switch than in visual flash mental math, so while you’re still getting used to mixed problems, practicing with listening arithmetic is one option. How it’s read aloud is explained in how listening arithmetic works.

Use running totals to find where you went wrong

When a mixed problem comes out wrong, the difference from the correct answer narrows down the cause. If you added a number you should have subtracted, your answer is bigger than the correct one by twice that number. For example, mistaking −23 for +23 puts the answer 46 off.

When checking answers, the flash mental math tool lists the numbers shown and the running totals, and if your answer matches one of the running totals, it tells you it equals the total up to that number. It also notes that if the difference equals one of the numbers shown, you may have counted it twice or missed it, and if the difference is a multiple of 10, you may have miscounted a carry or borrow. How to fix each cause is covered in detail in how to make fewer mental math mistakes.

FAQ

Why do I make mistakes in flash mental math when subtraction is mixed in?

Three things are added: switching between adding and subtracting, missing minus signs, and borrowing. Moving to mixed problems at the same speed as addition tends to fall apart, so it’s best to lengthen the seconds per number by about 1.5 times and then get used to it.

How do I subtract using the complement of 10?

When you can’t subtract from the ones place, take 1 from the tens place (that is, 10) and add the complement of 10 of the number you’re subtracting to the ones place. For 13 − 7, reduce the tens to get 3, then add 7’s complement, 3, to get 6.

Can the running total go negative partway through a mixed problem?

The flash mental math tool never gives a subtraction that would make the running total negative. The first number is always added, so you never run out of beads on the abacus either.

When should I start mixed addition and subtraction?

A good guide is when you can get 8 or more out of 10 right on addition at the same number of digits, at around 0.6 seconds per number. Starting when your abacus class begins mixed problems is also a good approach.