GCSE Maths Algebraic Fractions
GCSE Maths Algebraic Fractions – Introduction
Introduction
Maths Algebraic fractions are an important topic studied in GCSE Maths, specifically on the Higher Paper. This branch of mathematics deals with expressions that involve both numbers and variables, and maths algebraic fractions are an extension of these concepts. In this article, we will discuss why maths algebraic fractions are studied, their exclusivity to the Higher Paper, and the benefits they offer to students.
Why Are Maths Algebraic Fractions Studied?
Algebraic fractions hold significant relevance in various areas of mathematics and real-world applications. By studying maths algebraic fractions, students develop a deeper understanding of algebraic manipulation and problem-solving skills. These fractions are commonly encountered in algebraic equations, inequalities, and functions, making them essential for further mathematical studies.
Exclusivity to the Higher Paper
Maths Algebraic fractions are taught exclusively on the Higher Paper of the GCSE Maths syllabus. This paper is designed for students aiming to achieve higher grades and pursue advanced mathematics. By introducing maths algebraic fractions at this level, it challenges students to enhance their critical thinking and problem-solving abilities. The Higher Paper provides an opportunity for students to explore complex mathematical concepts and lay a strong foundation for further studies.
Benefits of Maths Algebraic Fractions
Simplifying Expressions: Maths Algebraic fractions allow for simplification of complex expressions by combining terms and reducing common factors. This simplification makes it easier to solve equations and work with maths algebraic functions.
Solving Equations: Maths Algebraic fractions are used extensively in solving equations with multiple variables. By manipulating fractions, students can isolate variables and find their values, leading to the solution of equations.
Understanding Function Behavior: Algebraic fractions help in analysing the behaviour of algebraic functions. By studying the properties of these fractions, students can identify asymptotes, intercepts, and other key features of functions.
Applications in Science and Engineering: Maths Algebraic fractions find applications in various scientific and engineering fields. They are used to model and solve real-world problems that involve proportional relationships, growth rates, and complex systems.
GCSE Maths Algebraic Fractions - Examples
\begin{aligned} &\begin{aligned} \frac{4(x+9)+5(x+2)}{20} & =5 \\ 4 x+36+5 x+10 & =100 \\ 9 x+46 & =100 \\ 9 x & =54 \end{aligned}\\ &x=6 \end{aligned}
\text { Solve } \frac{x-1}{2}+\frac{x+4}{5}=8
\begin{aligned} &\begin{aligned} \frac{5(x-1)+2(x+4)}{10} & =8 \\ 5 x-5+2 x+8 & =80 \\ 7 x+3 & =80 \\ 7 x & =77 \end{aligned}\\ &x=11 \end{aligned}
\text { Solve } \frac{x+5}{3}-\frac{x-2}{4}=3
\begin{aligned} &\begin{aligned} \frac{4(x+5)-3(x-2)}{12} & =3 \\ 4 x+20-3 x+6 & =36 \\ x+26 & =36 \end{aligned}\\ &x=10 \end{aligned}
\text { Solve } \frac{x+2}{8}+\frac{5-x}{3}=2
\begin{aligned} &\frac{3(x+2)+8(5-x)}{24}=2\\ &\begin{aligned} 3 x+6+40-8 x & =48 \\ 46-5 x & =48 \\ -5 x & =2 \\ x & =-\frac{2}{5} \end{aligned} \end{aligned}
GCSE Maths Algebraic Fractions - Further Examples
\begin{aligned} &\begin{aligned} \frac{3(x+3)+x+5}{(x+5)(x+3)} & =2 \\ 3 x+9+x+5 & =2\left(x^2+8 x+15\right) \\ 4 x+14 & =2 x^2+16 x+30 \\ 0 & =2 x^2+12 x+16 \\ 0 & =x^2+6 x+8 \\ 0 & =(x+4)(x+2) \end{aligned}\\ &x=-4 \quad x=-2 \end{aligned}
\text { Solve } \frac{2}{2 x+3}+\frac{3}{x-2}=1
\begin{aligned} & \frac{2(x-2)+3(2 x+3)}{(2 x+3)(x-2)}= 1 \\ & 2 x-4+6 x+9= 2 x^2-x-6 \\ & 8 x+5= 2 x^2-x-6 \\ & 0=2 x^2-9 x-11 \\ & 0=(2 x-11)(x+1) \\ & \quad x=\frac{11}{2} \quad x=-1 \end{aligned}
\text { Solve } \frac{3 x+1}{x+1}-\frac{1}{x+3}=4
\frac{(3 x+1)(x+3)-(x+1)}{(x+1)(x+3)}=4
\begin{aligned} 3 x^2+10 x+3-x-1= & 4\left(x^2+4 x+3\right) \\ 3 x^2+9 x+2= & 4 x^2+16 x+12 \\ 0= & x^2+7 x+10 \\ 0= & (x+5)(x+2) \\ & x=-5 \quad x=-2 \end{aligned}
GCSE Maths Algebraic Fractions – More Further Examples
\begin{aligned} \frac{7(x-3)-2(3 x+1)}{(3 x+1)(x-3)} & =3 \\ 7 x-21-6 x-2 & =3\left(3 x^2-8 x-3\right) \\ x-23 & =9 x^2-24 x-9 \\ 0 & =9 x^2-25 x+14 \\ 0 & =(9 x-7)(x-2) \\ x & =\frac{9}{7} \quad x=2 \end{aligned}
\text { Solve } \frac{6}{x+7}+\frac{2}{x-5}=\frac{2}{3}
\begin{aligned} \frac{6(x-5)+2(x+7)}{(x+7)(x-5)}= & \frac{2}{3} \\ 6 x-30+2 x+14 & =\frac{2}{3}\left(x^2+2 x-35\right) \\ 3(8 x-16) & =2 x^2+4 x-70 \\ 24 x-48 & =2 x^2+4 x-70 \\ 0 & =2 x^2-20 x-22 \\ 0 & =x^2-10 x-11 \\ 0 & =(x-11)(x+1) \quad x=-1 \end{aligned}
\text { Solve } \frac{5 x+2}{x+1}-\frac{x+8}{x+3}=2
\begin{aligned} &\begin{aligned} \frac{(5 x+2)(x+3)-(x+8)(x+1)}{(x+1)(x+3)} & =2 \\ 5 x^2+17 x+6-x^2-9 x-8 & =2\left(x^2+4 x+3\right) \\ 4 x^2+8 x-2 & =2 x^2+8 x+6 \\ 2 x^2-8 & =0 \\ x^2-4 & =0 \\ (x+2)(x-2) & =0 \end{aligned}\\ &x=-2 \quad x=2 \end{aligned}
\text { Solve } \frac{x}{2 x-1}+\frac{x-3}{2-x}=\frac{1}{4}
\begin{aligned} & \frac{x(2-x)+(x-3)(2 x-1)}{(2 x-1)(2-x)}=\frac{1}{4} \\ & 2 x-x^2+2 x^2-7 x+3=\frac{1}{4}\left(5 x-2-2 x^2\right) \\ & 4\left(x^2-5 x+3\right)=5 x-2-2 x^2 \\ & 4 x^2-20 x+12=5 x-2-2 x^2 \\ & 6 x^2-25 x+14=0 \\ &(3 x-2)(2 x-7)=0 \\ & x=\frac{2}{3} \quad x=\frac{7}{2} \end{aligned}
Conclusion
GCSE Maths Algebraic Fractions are an integral part of the Higher Paper curriculum. By studying algebraic fractions, students develop critical thinking skills, algebraic manipulation techniques, and problem-solving abilities. These fractions provide a foundation for further mathematical studies and have practical applications in various fields.