Counting by Weighing

Introduction

When a hardware shop sells 500 screws, nobody counts them out by hand. The assistant tips screws onto a scale until the reading reaches the right mass. Banks check bags of coins the same way. This trick — counting by weighing — works whenever every item has the same mass. It is exactly the trick chemists use to count atoms, which are far too small to count any other way.

Core explanation

The basic idea. If every item in a collection has the same mass, then:

number of items = total mass ÷ mass of one item

If one nail has a mass of 2.5 g, a pile of nails with a total mass of 500 g contains 500 ÷ 2.5 = 200 nails. The scale does the counting for you.

Working backwards. The same relationship can be rearranged. To get a chosen number of items, weigh out:

total mass = number of items × mass of one item

A shop wanting to sell 1000 nails weighs out 1000 × 2.5 g = 2500 g, or 2.5 kg.

Banks and coins. A British 1p coin has a mass of about 3.56 g. A bag containing exactly one pound in 1p coins (100 coins) should therefore weigh 356 g. Bank staff weigh bags instead of counting them; if the mass is wrong, the bag is recounted.

Items that are not all identical. Real nails vary slightly. Counting by weighing then uses the average mass of one item. As long as the sample is large, small differences average out and the count is very close to correct.

Comparing two kinds of item. Suppose large bolts are three times as heavy as small washers. Then 30 g of bolts and 10 g of washers contain the same number of items, because each bolt weighs three times as much as each washer. The key is that masses in the same ratio as the individual item masses always contain equal numbers of items.

From hardware to atoms. Atoms of the same element have (on average) the same mass, so they can be counted by weighing too. Relative atomic masses tell us how heavy atoms are compared with each other: carbon is 12, magnesium is 24, sulfur is 32. Therefore 12 g of carbon, 24 g of magnesium and 32 g of sulfur all contain the same number of atoms . This insight is the foundation of the mole, which fixes that number precisely.

Step-by-step reasoning

To count identical items by weighing:

1. Find the mass of one item (or the average mass of a known number of items). 2. Weigh the whole collection. 3. Make sure both masses are in the same unit. 4. Divide the total mass by the mass of one item to get the number of items. 5. Round to a whole number, because you cannot have part of an item.

Visual explanation

Imagine two jars on two balances. One holds golf balls, the other ping-pong balls. The golf-ball jar reads 460 g and the ping-pong jar 27 g. Each golf ball is 46 g and each ping-pong ball 2.7 g, so both jars hold exactly ten balls, even though their masses are very different.

Real-world analogy

A school cafeteria orders rice by the kilogram but thinks in portions. If one portion is 75 g, then a 15 kg sack provides 15 000 ÷ 75 = 200 portions. The cook never counts grains or scoops in advance; the mass alone tells her how many people she can feed.

Real-world example

Pharmacies use counting scales to fill prescriptions. The pharmacist places ten tablets on a digital counting balance, which records their average mass, then pours in more until the display shows the number required, such as 28 tablets. The same principle lets factories package millions of small parts quickly and accurately.

Why?

Why does dividing by the mass of one item give the number of items? Because mass adds up: the total mass of a collection is the sum of the individual masses. If all the masses are equal, the total is simply the number of items multiplied by the mass of each one, so division reverses this.

Common misconception

"Equal masses of different items contain equal numbers." Only equal masses of identical items do. A kilogram of feathers and a kilogram of marbles have the same mass but very different numbers of pieces, because each piece has a different mass.

Worked example

Question: A packet of paper clips has a mass of 180 g. Ten clips together have a mass of 4.5 g. How many clips are in the packet?

Reasoning: The mass of one clip is 4.5 ÷ 10 = 0.45 g. Number of clips = total mass ÷ mass of one clip = 180 ÷ 0.45 = 400.

Answer: 400 paper clips.

Quick check

1. One screw has a mass of 4 g. How many screws are in a 1 kg bag? Answer: 1 kg = 1000 g, and 1000 ÷ 4 = 250 screws.

Exam focus

Show the division clearly and check units: convert kilograms to grams before dividing. In atom questions, be ready to explain that masses in the ratio of the relative atomic masses contain equal numbers of atoms — for example, 12 g of carbon and 24 g of magnesium.

Advanced insight

Counting by weighing becomes more accurate as the sample grows, because random variations in individual masses partly cancel out. Atoms are an extreme case: even a milligram of an element contains around 10¹⁹ atoms or more, so the average mass is effectively exact. Isotopes cause individual atoms of one element to differ in mass, which is why relative atomic masses such as chlorine's 35.5 are averages.

Summary

Identical items can be counted by weighing: number = total mass ÷ mass of one item. For slightly varying items, the average mass is used. Masses in the same ratio as the individual item masses contain equal numbers of items. Because relative atomic masses compare atom masses, 12 g of carbon and 24 g of magnesium contain the same number of atoms — the idea behind the mole.

Practice questions

1. A washer has a mass of 1.2 g. How many washers are in a bag of mass 600 g? Answer: 600 ÷ 1.2 = 500 washers. 2. What total mass of 1p coins (3.56 g each) makes £5? Answer: £5 is 500 coins, so 500 × 3.56 = 1780 g, or 1.78 kg. 3. Each large bead is four times as heavy as each small bead. What mass of small beads contains the same number of beads as 80 g of large beads? Answer: 80 ÷ 4 = 20 g of small beads. 4. Using relative atomic masses C = 12 and S = 32, what mass of sulfur contains the same number of atoms as 6 g of carbon? Answer: 6 g is half of 12 g, so half of 32 g of sulfur is needed: 16 g.