Physical Quantities, Part 4 — Errors and How They Add Up

Welcome to the last stop in our Physical Quantities series. We have come a long way — we met physical quantities, sorted scalars from vectors, and got units straight. Now for the part JEE loves most: errors.

If you skipped ahead, here is the trail so far:

• Part 1 introduced physical quantities.
• Part 2 split the world into scalars and vectors.
• Part 3 was all about units.

Why errors matter so much

Physics is an experimental science. And anything to do with experiments shows up on JEE, NEET and the rest — year after year, without fail. The good news? These questions are easy marks once you know the trick. So don't leave them on the table.

Where errors come from

Every experiment starts with a measurement. And no measurement is perfect. Two things hold it back:

• The instrument. Could you measure the thickness of a needle with an ordinary ruler? Of course not. Every tool has a limit.
• The person. A tired hand reading a scale at 11pm will sometimes write 6 where the truth is 5. We all do it.

So no experiment is ever 100% error-free. The honest move is not to pretend otherwise — it is to measure how wrong you might be.

Three ways to name an error

Say you measure a length and call it $x$, but the true value could be off by an amount $\Delta x$. That little $\Delta x$ is your absolute error — same units as the quantity itself.

Absolute error alone doesn't tell you much. An error of 1 cm is tiny on a cricket pitch and huge on a matchstick. So we compare it to the value:

$\text{Relative error} = \frac{\Delta x}{x}$

Multiply that by 100 and you get the percentage error:

$\text{Percentage error} = \frac{\Delta x}{x} \times 100\%$

This is the number JEE almost always wants. Keep it in front of you.

The one rule: errors always add

Before any formula, hold on to two ideas:

• "Error" means the maximum possible error — the worst case.
• Errors always add up. Two wrongs never cancel to make a right. So you never subtract one error from another, even when the formula has a minus sign in it.

That second line is the whole secret. Let me show you.

Sums and differences

When quantities are added or subtracted, their absolute errors add:

$\text{if } X = A + B \text{ or } X = A - B, \quad \text{then} \quad \Delta X = \Delta A + \Delta B$

Example 1. $X = A + B$. The percentage errors in $A$ and $B$ are $1\%$ and $3\%$. Find the percentage error in $X$.

It's an addition, so the errors add: $(1 + 3)\% = 4\%$.

Example 2. Now $X = A - B$, same errors of $1\%$ and $3\%$. Find the error in $X$.

Tempted to do $3 - 1 = 2\%$? Don't. Remember — error always adds. The answer is still $4\%$. The minus sign in the formula changes nothing about the error.

Products, divisions and powers

For anything multiplied, divided, or raised to a power, switch to percentage (relative) errors and use the power trick.

The rule: for

$Y = M^a \, L^b \, T^{-c}$

the relative errors add, each scaled by its power — and you treat every power as positive, even the negative one:

$\frac{\Delta Y}{Y} = a\frac{\Delta M}{M} + b\frac{\Delta L}{L} + c\frac{\Delta T}{T}$

Two habits make this painless:

1. Ignore constants. Numbers like $\tfrac{4}{3}$ or $\pi$ carry no error, so drop them.
2. Multiply each percentage error by its power, and throw away any minus signs on the powers before you add.

Worked example. The radius of a sphere is measured with a $1\%$ error. What is the percentage error in its volume?

Start with the formula:

$V = \frac{4}{3}\pi r^3$

The $\tfrac{4}{3}$ and $\pi$ are constants — gone. Radius carries a power of 3, so:

$\frac{\Delta V}{V} = 3 \times \frac{\Delta r}{r} = 3 \times 1\% = 3\%$

A $1\%$ slip in radius becomes a $3\%$ slip in volume. That's the power at work — and exactly why high-power quantities need careful measuring.

Your turn. A quantity is $Y = M^a L^b T^{-c}$. The percentage errors in $M$, $L$, $T$ are $\alpha\%$, $\beta\%$, $\gamma\%$. What is the total percentage error in $Y$?

Check: Multiply each error by its power, drop the minus sign on $c$, and add them all: $(a\alpha + b\beta + c\gamma)\%$. Never subtract — errors always add.

The whole chapter in four lines

• Relative error is $\dfrac{\Delta x}{x}$; times 100 gives percentage error.
• Sums and differences: absolute errors add — even across a minus sign.
• Products, divisions, powers: percentage errors add, each times its power, signs ignored.
• When in doubt, add. Errors never cancel.

These questions are short, safe marks. With this one rule, you have them all. That wraps our Physical Quantities series — quantities, vectors, units, and now errors. Go find an old experiment problem and try the rule on it.