Mathematics · Expressions, equations and inequalities · Grade 7

Algebraic fractions

One pipe fills a pool in 4 hours, another in 12 hours. How long if both are open?

Open the experiment

The situation

Two pipes pour water into a pool: the first alone fills it in 4 hours, the second in 12 hours. How long will the pool take to fill if we open both together? In one hour the first pipe pours 1/4 of the pool and the second 1/12 — we need to add fractions. If we call the pipes' times a and b, we get 1/a + 1/b: a fraction with letters in the numerator and denominator. How do the rules for numerical fractions carry over to fractions with letters?

Topic

Algebraic fractions: simplifying, the common denominator, adding, subtracting, multiplying, dividing, rational expressions. A pool with two pipes: 1/a + 1/b — fractions of time; the rules of numerical fractions carry over to fractions with letters

Key ideas

• Cancelling
• Common denominator
• Fractional equation

Formulas

Algebraic fraction: the numerator $P$ and the denominator $Q$ are polynomials. For values that make the denominator zero, the fraction is undefined.

Cancelling: we factorise the numerator and the denominator and cancel the common factor ($x\ne 0$, $x\ne -2$).

Common denominator: we multiply each fraction by an extra factor to bring them to the same denominator; together the two pipes fill the pool in $t={\displaystyle \frac{ab}{a+b}}$ hours.

Multiplying — numerator by numerator, denominator by denominator; dividing — multiply by the second fraction turned upside down.

Theory

Algebraic fractions and the domain

An expression whose numerator and denominator are polynomials is an algebraic fraction: ${\displaystyle \frac{x}{x-2}}$, ${\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}$. If we give the letters numbers, the fraction becomes a number: at $x=5$ it is ${\displaystyle \frac{5}{3}}$. But at $x=2$ the denominator is $0$ — we cannot divide by zero, and the fraction is undefined. The values of the letters for which the fraction makes sense form its domain: $x\ne 2$. Before every transformation we write down the values that make the denominator zero.

The basic property and cancelling

As with numerical fractions: if we multiply or divide the numerator and the denominator by the same non-zero polynomial, the fraction does not change. To cancel, we first factorise the numerator and the denominator — the common factor cannot be seen while the brackets are expanded: ${\displaystyle \frac{x^2-4}{x^2+2x}}={\displaystyle \frac{(x-2)(x+2)}{x(x+2)}}={\displaystyle \frac{x-2}{x}}$. The cancelled fraction equals the original only when $x\ne -2$ (the original was undefined at $x=-2$). A common mistake: "cancelling" the $x$ in ${\displaystyle \frac{x+2}{x+3}}$ — here $x$ is not a factor but a term.

Common denominator

To add fractions with different denominators we bring them to a common denominator — usually the lowest common multiple. For the numbers $4$ and $12$ the LCM is $12$; for polynomials we factorise each denominator and take each factor once, to its highest power: for $x^2-1=(x-1)(x+1)$ and $x+1$ the common denominator is $(x-1)(x+1)$. Each fraction is multiplied by its extra factor. In the pool: ${\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{12}}={\displaystyle \frac{3}{12}}+{\displaystyle \frac{1}{12}}={\displaystyle \frac{4}{12}}={\displaystyle \frac{1}{3}}$ — a third of the pool in one hour, so it fills in $3$ hours.

Adding and subtracting

For fractions with the same denominator, the numerators are added (subtracted) and the denominator stays. If the denominators differ, we first find a common denominator: ${\displaystyle \frac{1}{x-1}}-{\displaystyle \frac{1}{x+1}}={\displaystyle \frac{(x+1)-(x-1)}{(x-1)(x+1)}}={\displaystyle \frac{2}{x^2-1}}$. When subtracting, the second numerator must be put in brackets: $-(x-1)=-x+1$. Always check the result: at $x=4$ the left-hand side is ${\displaystyle \frac{1}{3}}-{\displaystyle \frac{1}{5}}={\displaystyle \frac{2}{15}}$, and the right-hand side is ${\displaystyle \frac{2}{15}}$.

Multiplying, dividing, identity transformations

When multiplying, numerator is multiplied by numerator and denominator by denominator, but it is better to factorise and cancel first: ${\displaystyle \frac{x^2-4}{3x}}\cdot {\displaystyle \frac{6x^2}{x+2}}={\displaystyle \frac{(x-2)(x+2)\cdot 6x^2}{3x(x+2)}}=2x(x-2)$. When dividing, the second fraction is turned upside down: ${\displaystyle \frac{P}{Q}}:{\displaystyle \frac{R}{S}}={\displaystyle \frac{P}{Q}}\cdot {\displaystyle \frac{S}{R}}$. An expression with several operations is simplified step by step — this is an identity transformation: the result equals the original expression at every point of the domain, for example $({\displaystyle \frac{1}{x}}+{\displaystyle \frac{1}{y}}):{\displaystyle \frac{x+y}{xy}}=1$.

Equations with fractions and a note on the model

An equation with an unknown in the denominator is a fractional equation: ${\displaystyle \frac{3}{x-1}}={\displaystyle \frac{2}{x+1}}$. First the domain ($x\ne 1$, $x\ne -1$), then we multiply both sides by the common denominator: $3(x+1)=2(x-1)$, $x=-5$ — it lies in the domain, so it is a root. If the number we find makes a denominator zero, it is an extraneous root and is discarded: ${\displaystyle \frac{x}{x-3}}={\displaystyle \frac{3}{x-3}}$ gives $x=3$, but $x\ne 3$ — the equation has no roots. In the experiment, fractions are checked as decimals to two decimal places or as simplified common fractions; in the pool model the pipes are assumed to pour at a steady rate.

Did you know?

• Writing a fraction with a horizontal bar appears in the works of al-Hassar, a mathematician who worked in the Maghreb in the 12th century; it was brought to Europe by Fibonacci in his book “Liber abaci” (1202).
• The pool formula works in physics too: for two conductors connected in parallel, the total resistance follows from ${\displaystyle \frac{1}{R}}={\displaystyle \frac{1}{R_1}}+{\displaystyle \frac{1}{R_2}}$ as $R={\displaystyle \frac{R_1{R}_2}{R_1+R_2}}$ — just like the combined time of two pipes.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

• Expressions: from numbers to letters — Grade 7
• Monomials and polynomials — Grade 7
• Special products (algebraic identities) — Grade 7
• The linear equation: the balance — Grade 7
• Systems of equations — Grade 7
• Inequalities: ray and interval — Grade 8
• The quadratic equation: the parabola — Grade 8