2.9.2 Equations & Problem solving
In mathematics, problem solving entails using several stages across a variety of topics to answer a question.
In this set of notes all the problems will involve equations.
These could be linear equations, quadratic equations, simultaneous equations, or other simple equations.
Worked example:
Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.
Every small cake costs \( $x \) and every large cake costs \( $(2x+1) \).
Petra spends \( $20 \) on small coffee cakes and \( $10 \) on large coffee cakes.
The total number of cakes is \( 45 \).
Write an equation in terms of \( x \).
Solve this equation to find the cost of a small coffee cake.
Show all your working.
Let one small coffee cake be \( x \) and one large coffee cake be \( 2x+1 \)
Let the number of small coffee cakes Petra bought be \( a \) and the number of large coffee cakes be \( b \).
Then total number of coffee cakes that Petra bought will be
\( a+b=45 \) \( (1) \)
Number of small coffee cakes that can be bought for \( $ \ 20 \)
\( ax=20 \)
\( a=\frac{20}{x} \)
Number of large coffee cakes that can be bought for \( $10 \)
\( b(2x+1)=10 \)
\( b=\frac{10}{2x+1} \)
Substitute \( a=\frac{20}{x} \), \( b=\frac{10}{2x+1} \) in equation \( (1) \)
\( \frac{20}{x}+\frac{10}{2x+1}=45 \)
Multiply both sides by the common denominator which is \( x(2x+1) \)
\( \frac{20}{x}\times x(2x+1)+\frac{10}{2x+1}\times x(2x+1)=45\times x(2x+1) \)
Cancel the common factors
\( 20\times (2x+1)+10x=45x\times (2x+1) \)
Expand the brackets and collect like terms
\( 40x+20+10x=90x^2+45x \)
\( 90x^2-5x-20=0 \)
\( 18x^2-x-4=0 \)
Use the quadratic formula
\( a=18 \), \( b=-1 \), \( c=-4 \)
\( x=\frac {-b\pm \sqrt{b^2-4ac}}{2a} \)
Substitute the values in the formula
\( x=\frac{-(-1)\pm \sqrt{(-1)^2-4\times18 \times(-4)}}{2×18} \)
\( x=\frac{1\pm \sqrt{289}}{36} \)
\( x=\frac {1\pm 17}{36} \)
\( x=\frac{1+17}{36} \), \( x=\frac{1-17}{36} \)
\( x=\frac{1}{2} \), \( x=-\frac{4}{9} \)
\( x=\frac{1}{2} \) because price must be positive. So one small coffee cake costs \( $0.50 \).
Test yourself:
Question 1:
Alan invests \( $200 \) at a rate of \( r \)% per year compound interest.
After \( 2 \) years the value of his investment is \( $206.46 \) .
Show that \( r^2+200r-323=0 \).
[3]
Solve the equation \( r^2+200r-323=0 \) to find the rate of interest.
Show all your working and give your answer correct to \( 2 \) decimal places.
[3]
Question 2:
\( A \), \( B \) and \( C \) lie on the circle, center \( O \).
Angle \( AOC=3x+22° \) and angle \( ABC=5x° \).
Find the value of \( x \) .
[4]
Question 3:
In a shop, the price of a monthly magazine is \( $ \ m \) and the price of a weekly magazine is \( $ \ (m-0.75) \).
One day, the shop receives
\( $ \ 168 \) from selling monthly magazines
\( $ \ 207 \) from selling weekly magazines
The total number of these magazines sold during this day is \( 100 \).
Show that \( 50m^2-225m+63=0 \)
[3]
Find the price of a monthly magazine.
Show all your working.
[3]
Question 4:
At a football match, the price of an adult ticket is \( $ \ x \) and the price of a child ticket is \( $ \ (x-2.50) \).
There are \( 18500 \) adults and \( 2400 \) children attending the football match.
The total amount paid for the tickets is \( $ \ 320040 \).
Find the price of an adult ticket.
[4]
Question 5:
Petra records the score in each test she takes.
The mean of the first \( n \) scores is \( x \).
The mean of the first \( n-1 \)scores is \( x+1 \).
Find the \( n^{th} \) score in terms of \( n \) and \( x \).
Give your answer in its simplest form.
[3]