Checking Answers with Units

Introduction

A calculator never argues. If you type the numbers in the wrong order, it will happily give you a precise-looking answer that is completely wrong. Units are your first line of defence. If you write units next to every number and carry them through the working, a mistake in the equation almost always shows up as a strange unit in the answer. Combined with a quick check of the size of the number, this simple habit catches most errors before they cost marks.

Core explanation

Carry units through every step. When you substitute numbers into an equation, write the unit next to each value and treat the units algebraically, as in dimensional analysis. At the end, the units combine to give the unit of the answer.

For example, density = mass ÷ volume. With a mass of 54 g and a volume of 20 cm³:

density = 54 g ÷ 20 cm³ = 2.7 g/cm³

The unit g/cm³ is a genuine density unit, so the equation was arranged correctly.

A wrong unit reveals a wrong equation. Suppose you mistakenly calculate volume ÷ mass. You get 20 cm³ ÷ 54 g = 0.37 cm³/g. That is not a unit of density. Similarly, if you want a volume from density and mass and you multiply them instead of dividing, the unit comes out as g²/cm³ — clearly nonsense. The unit tells you the equation is upside down or the operation is wrong.

Use consistent units. Before combining numbers, make sure they are on the same scale. Mixing a volume in dm³ with one in cm³, or a mass in kg with one in g, gives answers that are wrong by a factor of 1000. Convert everything to matching units first.

Units cannot be added unless they match. You can add 25 cm³ to 10 cm³, but adding 25 cm³ to 10 g has no meaning. If your working asks you to add or subtract quantities with different units, something has gone wrong.

Check the size as well as the unit. A correct unit does not guarantee a correct number. Ask whether the answer is sensible: a density of 270 g/cm³ for a metal is impossible, because the densest elements are only about 22 g/cm³. A rough mental estimate, rounding each number to one figure, shows whether the answer is roughly the right size.

Step-by-step reasoning

1. Write every value with its unit. 2. Convert values to consistent units. 3. Substitute into the equation and combine the units algebraically. 4. Compare the final unit with the quantity you want. 5. Estimate the answer roughly and compare it with your calculated value.

Visual explanation

Imagine a checklist beside every answer with two boxes: "unit correct?" and "size sensible?". Only when both boxes are ticked is the answer accepted. A cross in either box sends you back to the working.

Real-world analogy

Units are like the labels on jars in a kitchen. If a recipe tells you to add salt and you reach for a jar labelled "sugar", the label warns you before you spoil the cake. A calculation without units is like a kitchen full of unlabelled jars.

Real-world example

In 1999 NASA lost the Mars Climate Orbiter spacecraft because one engineering team supplied thruster data in pound-force seconds while the navigation software expected newton seconds. The numbers looked reasonable, but the units did not match, and the craft flew far too close to Mars.

Why?

Why does a wrong equation usually give a wrong unit? Each physical quantity has a characteristic combination of units. Dividing instead of multiplying, or turning a fraction upside down, changes that combination, so the result cannot have the unit of the quantity you were aiming for.

Common misconception

"If the unit is right, the answer must be right." A correct unit shows that the equation has the right form, but it cannot catch arithmetic slips, a missing factor of ten or a misread value. Always check the size of the answer too.

Worked example

Question: A student calculates the volume of 39.5 g of ethanol (density 0.79 g/cm³) as 39.5 × 0.79 = 31.2. Check the answer using units.

Reasoning: Units: g × g/cm³ = g²/cm³, which is not a volume unit. The correct rearrangement is volume = mass ÷ density: 39.5 g ÷ 0.79 g/cm³ = 50 cm³. Here g cancels, leaving cm³. The size is sensible because ethanol is less dense than water, so 39.5 g should occupy more than 39.5 cm³.

Answer: The student's method is wrong; the volume is 50 cm³.

Quick check

1. A calculation for density gives an answer in cm³/g. What does this suggest? Answer: The equation has been inverted: volume was divided by mass instead of mass by volume.

Exam focus

Always include units in your final answer; many mark schemes withhold a mark for a missing or wrong unit. When asked to "suggest" whether an answer is reasonable, comment on both the unit and the size of the number.

Advanced insight

Scientists sometimes use unit checks as a design tool. Before building a model, they write the units of every quantity and see which combinations can produce the required unit. This approach, called dimensional reasoning, can reveal the form of a relationship even before any experiment is carried out.

Summary

Write units next to every number and carry them through the working. The final unit shows whether the equation was set up correctly: a strange unit points to an inverted fraction or a wrong operation. Use consistent units, never add quantities with different units, and check that the size of the answer is sensible.

Practice questions

1. A student adds 2.0 kg to 350 g and writes 352. What has gone wrong, and what is the correct total? Answer: The units were not made consistent; 2.0 kg = 2000 g, so the total is 2350 g (2.35 kg). 2. What unit results from multiplying a density in g/cm³ by a volume in cm³, and what quantity is this? Answer: Grams, because cm³ cancels; the quantity is mass. 3. A student calculates the density of a rock as 250 g/cm³. Explain why this is unlikely. Answer: No known solid is that dense; the densest elements are about 22 g/cm³, so there is probably an error such as a misplaced decimal point. 4. Why can a correct unit still accompany a wrong answer? Answer: Units check only the form of the equation, not arithmetic mistakes or misread values, so the number can still be wrong.