Asking an assistant to show its arithmetic

TL;DR:

• Asking for each line of working turns a number you must trust into steps you can check.

• Recompute each line yourself, and then work backwards from the answer to test it.

• A wrong answer usually hides in one line, and showing the work shows you which.

Ask for a number and you get a number. Whether that number is right is a separate matter, and a bare number gives you no way to tell. Ask for the working, and the same answer turns into a few short lines you can check in under a minute. This is the cheapest accuracy upgrade there is.

Here is a percentage problem checked line by line, including what a wrong answer looks like when the work is on show.

The problem

Say a jacket costs $80. It is marked 25% off, and then a sales tax of 8% is added to the reduced price. What is the final price?

Walk-through: a percentage problem checked line by line

Suppose you ask for the answer and get "$66.40". Looks plausible. Now ask for the working.

The working that gives $66.40

1. Net change: −25% + 8% = −17%.

2. Price: 80 × (1 − 0.17) = 80 × 0.83.

3. Result: 66.40.

Read line 1. It adds the two percentages together. That is the error: the tax applies to the reduced price, not the original, so the two cannot simply be added. Showing the work made the mistake visible in one line.

The correct working

1. Discount: 80 × 0.75 = 60.00. Taking 25% off means keeping 75%.

2. Tax on the reduced price: 60.00 × 1.08 = 64.80.

3. Final price: $64.80.

Check line 1: 25% of 80 is 20, and 80 − 20 = 60. Check line 2: 8% of 60 is 4.80, and 60 + 4.80 = 64.80. Both agree.

Work backwards

Undo the steps in reverse. 64.80 ÷ 1.08 = 60.00, and 60.00 ÷ 0.75 = 80.00. You arrive back at the original $80, so the answer is consistent.

The wrong answer, 66.40, fails the reverse test: 66.40 ÷ 1.08 ≈ 61.48, and 61.48 ÷ 0.75 ≈ 81.98, which is not 80.

What to ask for

• "Show your working, one line per step."

• "Write the units next to each number."

• "State what each step does in a few words."

• "Check the answer by working backwards."

How to check each line

1. Is the operation the right one? A discount multiplies by one minus the rate, as in 1 − 0.25 = 0.75. Tax multiplies by one plus the rate, as in 1 + 0.08 = 1.08.

2. Does the line follow from the one before? Each starts from the previous result.

3. Is the arithmetic right? Redo it on a calculator.

4. Do the units match? Money stays in money, and a rate stays a rate.

5. Is the size sensible? A discount then a smaller tax should land below the original $80. It does: $64.80.

Tip: The order matters. A discount followed by tax gives 64.80, and so does tax followed by the discount, since 80 × 1.08 × 0.75 = 64.80. But adding the percentages together does not, which is why that shortcut fails.

Why it helps with other maths too

The same trick works for fractions, unit conversions, interest, averages and anything with more than one step. Each extra line is a place a mistake can be seen. People who get into the habit of writing their working catch their own errors much more often, because the working is there to be reread. An assistant that writes its steps hands you that same chance.

If you are checking a figure that matters, like a bill, a budget or a grade calculation, do the reverse step as a last check. Working backwards takes seconds and catches errors the forward pass can repeat.

Common slips to watch for

A few mistakes come up again and again in percentage work. Treating 25% as 25 instead of 0.25. Taking the percentage of the wrong base. Applying a discount after tax in one place and before in another. Rounding too early, which slowly drifts the answer. When you read a line of working, ask yourself which of these could be hiding in it.

What showing the work does not do

A neat set of steps can still contain a wrong step that looks reasonable. The check is on you: read each line with fresh eyes, and recompute. Showing the work makes errors findable. It does not make them impossible.

Privacy and control

DuckyHelper can work a problem out with you on the blackboard, line by line, and it sees your screen only while the pill is open and for that answer. Whatever tool you use, the habit of reading each line is yours.

Try it yourself

Think of a price you saw this week with a discount. Work out the final price yourself first, then ask an assistant to show its steps. If the two agree, you have confirmed it. If not, find the line where they split.

Frequently asked questions

Why does adding percentages go wrong?

Because each percentage applies to a different base. The discount applies to 80, the tax to the reduced price.

Do I need to check everything this way?

Check numbers that matter. For a quick estimate, a glance may do.

What if the steps look right but the answer is wrong?

Recompute each line. A slip often hides in a multiplication.