Question

In a Mathematics quiz,$30$prizes consisting of $1\mathrm{st}$ and $2\mathrm{nd}$ prizes only are to be given.

$1\mathrm{st}$ and $2\mathrm{nd}$ prizes are worth $₹2000$ and $₹1000$ respectively.

If the total prize money is $₹52,000$then show that:

(a) If $1\mathrm{st}$ prizes are$\mathrm{x}$ in number the number of $2\mathrm{nd}$ prizes are__________.

(b)The total value of prizes in terms of$\mathrm{x}$ are__________.

(c) The equation formed is________.

(d) The solution of the equation is_______.

(e) The number of $1\mathrm{st}$ prizes are______________ and the number of $2\mathrm{nd}$ prizes are__________.

Answer 1

Step 1 : Find the required value for $2^{nd}$ prize

Given: Total prizes $=30$, $1^{st}$ prize money is $₹2000$, $2^{nd}$ prize money is $₹1000$ and the total prize money is $₹52,000$.

Let the numbers of $1^{st}$ prizes are$x$

Then the numbers of $2^{nd}$ prizes be $(30-x)$

Step 2: Total value of prizes in terms of $x$

Total value $=2000\times x+1000(30-x)$

$2000x+30000-1000x=52000$

$1000x=22000$

Step 3: Solving the equation formed

$$\begin{gathered} 1000x=22000 \\ \Rightarrow x=\frac{22000}{1000} \\ \Rightarrow x=22 \end{gathered}$$

Therefore,

$$\begin{gathered} (30-x)=30-22 \\ \Rightarrow (30-x)=8 \end{gathered}$$

Hence,

(a) If $1st$ prizes are$x$ in number the number of $2nd$ prizes are $\underline{(30-x)}$ .

(b)The total value of prizes in terms of$x$ are $\underline{2000\left(x\right)+1000(30-x)}$ .

(c) The equation formed is$\underline{1000x+30000=52000}$.

(d) The solution of the equation is$\underline{22}$.

(e) The number of $1st$ prizes are $\underline{22}$ and the number of $2nd$ prizes are $\underline{8}$.