The library

A certain number of people agreed to contribute equally to buy books worth $12000 for a school library. Five people pulled out, so others agreed to contribute an extra $100 each. Their contribution enabled them to buy books worth $ 2000 more than they originally expected. Write down:

a) an expression of how much each was originally to contribute.
b) two distinct expressions of how much each contributed after the five pulled out.
c) Calculate the value of x

Final Answer:

a =  300 USD
x =  40

Step-by-step explanation:

$$\begin{gathered} 12000=x\cdot a \\ 12000+2000=(x-5)\cdot (a+100) \\ 12000=x\cdot a-5\cdot a+100x-500 \\ 14000=12000-5\cdot a+100x-500 \\ 2500=-5\cdot a+100x \\ 500=20x-a \\ a=20x-500 \\ 12000=x\cdot a \\ 12000=x\cdot (20\cdot x-500) \\ 20x^2-500x-12000=0 \\ {x}^2-25x-600=0 \\ {x}^2-25x-600=0 \\ a=1;b=-25;c=-600 \\ D=b^2-4ac=25^2-4\cdot 1\cdot (-600)=3025 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{25\pm \sqrt{3025}}{2} \\ {x}_{1,2}=\frac{25\pm 55}{2} \\ {x}_{1,2}=12.5\pm 27.5 \\ {x}_1=40 \\ {x}_2=-15 \\ x=x_1=40 \\ a=20\cdot x-500=20\cdot 40-500=300 \\ \text{ Verifying Solution: } \\ b=a+100=300+100=400\text{USD} \\ {t}_1=x\cdot a=40\cdot 300=12000\text{USD} \\ {t}_2=(x-5)\cdot b=(40-5)\cdot 400=14000\text{USD} \end{gathered}$$

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