Number and Algebraic Foundations
Arithmetic of Fractions: a Working Reference
Building up, reducing, adding, subtracting, multiplying and dividing fractions, why cross-multiplication is the wrong habit, and how compound fractions are cleared.
What this page covers
- Build a fraction to a required denominator and reduce it to lowest terms
- Add and subtract fractions using a common denominator rather than cross-multiplying
- Divide by inverting and multiplying, and say why that works
- Simplify a compound fraction with fractions in its numerator or denominator
Terminology
- Numerator
- The top of the fraction — how many parts are taken
- Denominator
- The bottom — how many parts the whole is divided into
- Proper fraction
- Numerator smaller than denominator, e.g. 25
- Improper fraction
- Numerator at least as large as the denominator, e.g. 75
- Mixed number
- An integer plus a proper fraction, e.g. 125
- Equivalent fractions
- Different-looking fractions of equal value, e.g. 25 and 410
- Lowest terms
- Numerator and denominator share no common factor above 1
Every integer is already a fraction: the source notes point out that 6 = 61. That observation is what allows an integer to be brought into a fraction calculation without any special case.
For algebraic work, improper fractions are almost always more convenient than mixed numbers. 75 multiplies and divides cleanly; 125 has to be converted first, and in an algebraic expression it reads as a product rather than a sum.
Building up and reducing
These are the same operation run in opposite directions, and both rest on one identity.
This works because akbk = ab · kk = ab · 1. Multiplying by 1 is exactly what leaves a value unchanged, which is why the identity element matters here.
Building up
Building up raises a fraction to a required denominator, which is the whole preparation step for addition.
Reducing to lowest terms
Reducing runs the identity backwards: identify a factor common to numerator and denominator and remove it.
Worked example — numeric and algebraic
The algebraic case works identically once the powers are written out as repeated factors:
Cancelling one x and three ys leaves x on top and 2y2 below. Writing the powers out in full the first few times makes it obvious which factors survive.
Cancellation removes a factor, never a term. In x + 2x nothing cancels, because the numerator is x + 2, not x times something. Testing with a number settles it: at x = 2 the fraction is 42 = 2, not 22 = 1.
Adding and subtracting
Fractions with the same denominator add directly, because they count the same size of part.
Different denominators must first be made the same. That is all a common denominator is: a size of part that both fractions can be expressed in.
- Find a common denominator. The product of the two always works; the lowest common multiple keeps the arithmetic smaller.
- Build each fraction up to that denominator using the fundamental identity.
- Add or subtract the numerators and keep the denominator.
- Reduce the result to lowest terms.
Worked example — 23 + 415
Only the first fraction needed building, because 15 is already a multiple of 3.
Worked example — 710 - 56
The source builds both to the product 30:
Note the sign. Subtracting the larger fraction gives a negative result, and the minus sign belongs to the whole fraction: -215 = -215.
Worked example — 1724 - 340
Here the lowest common denominator is 120, not 24 · 40 = 960, which saves a substantial reduction.
The notes state, in capitals and underlined, do not cross multiply. The warning is aimed at addition and subtraction. Cross-multiplication is a legitimate move when solving ab = cd, but it is not how fractions are combined, and treating it as such produces reliable errors once compound fractions or three-term sums appear.
Multiplying and dividing
Cancelling common factors before multiplying keeps the numbers small and usually removes the need to reduce afterwards.
The rule is the multiplicative-inverse law in disguise. Dividing by cd means multiplying by its reciprocal, and the reciprocal of cd is dc, because cd · dc = 1.
Worked example — a compound fraction
The source poses 35 divided by 47 written as a single stacked fraction, and clears it two ways.
By inverting the divisor:
By building both parts to a common denominator first, which is the method that generalises to algebraic compound fractions:
2120 = 1.05, and 0.6 ÷ 0.5714… ≈ 1.05. The two routes agree.
The second method — multiply the whole compound fraction, top and bottom, by the common denominator of the inner fractions — is worth learning even though it looks longer. It handles cases such as ab - c + bb + caa2 - b2 where inverting is not straightforward, and the source notes use exactly that manoeuvre on page 9.
Fractions with negative powers
Negative exponents produce fractions, and the source works a combined example that exercises the whole toolkit at once.
Worked example — x-3y5 - 3x-4y6
Rewrite the negative powers as fractions:
Build to the common denominator x4:
Factoring the numerator at the end is worth the extra line: it exposes the zero at x = 3y, which the unfactored form hides.
Quick reference
| Operation | Rule | Common denominator needed? |
|---|---|---|
| Build up | ab = akbk, k ≠ 0 | — |
| Reduce | Divide top and bottom by a common factor | — |
| Add | ac + bc = a + bc | Yes |
| Subtract | ac - bc = a - bc | Yes |
| Multiply | ab · cd = acbd | No |
| Divide | ab ÷ cd = ab · dc | No |
The one asymmetry worth memorising: addition and subtraction need a common denominator, multiplication and division do not. Nearly every fraction error traces back to that distinction being blurred.
Common mistakes
| Mistake | Correct result | Fast check |
|---|---|---|
| 23 + 14 = 37 | 1112 | Adding tops and bottoms would make 12 + 12 = 24 = 12, which is plainly false. |
| x + 2x = 2 | No simplification | At x = 2 the value is 2; at x = 1 it is 3. It is not constant. |
| 1a + 1b = 1a + b | a + bab | Take a = b = 1: the left is 2, the right is 12. |
| Inverting the wrong fraction when dividing | Invert the divisor only | Estimate: dividing by a number under 1 must make the result larger. |
| Losing the sign in -430 | -215 | Carry the minus sign in front of the whole fraction from the start. |
| Cross-multiplying to add | Build to a common denominator | Cross-multiplication solves an equation between two fractions; it does not combine them. |
Frequently asked questions
Why is cross-multiplication wrong for addition?
It is not wrong for solving an equation of the form a/b = c/d, which is where it belongs. It is wrong as a way of adding, because a/b + c/d is not (ad + bc)/(bd) arrived at by 'crossing' — it is that expression arrived at by building both fractions to the common denominator bd. Students who learn the shortcut without the reason apply it to subtraction and to compound fractions, where it fails.
Do I have to use the lowest common denominator?
No. Any common denominator gives a correct answer; the lowest one just keeps the numbers small and often saves a reduction at the end. 710 - 56 can be done over 60 or over 30; both are right.
Why does dividing by a fraction mean multiplying by its reciprocal?
Because dividing by b means multiplying by b-1, and the reciprocal of 47 is 74. So 35 ÷ 47 is 35 · 74. The rule is the multiplicative-inverse law, not a separate trick.
Can I cancel across a plus sign?
No. In x + 2x the x terms cannot be cancelled, because the numerator is a sum, not a product. Cancellation removes a factor common to the whole numerator and the whole denominator. This is one of the most persistent errors in algebra.
Source. Handwritten teaching notes, Week 2, pages 6-8, together with the typed supplementary notes on fractions.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.