Question

A sum of $2700$ is to be given in the form of 63 prizes. If the prize is of either $100$ or $25$, find the number of prizes of each type.

Answer 1

Hint: We assume a number of $100$ prizes and $25$ prizes as two different variables and form two linear equations in two variables using the given equations. Find the value of variables using the substitution method.

Complete step-by-step solution:
Let number of $100$ prizes be ‘x’ and number of $25$ prizes be ‘y’
We know total number of prizes is 63; we can form an equation for sum of each type of prizes equal to given number of prizes
\[ \Rightarrow x + y = 63\].................… (1)
Now we know there are ‘x’ prizes of $100$ each
\[ \Rightarrow \]Total prize money for $100$ prizes\[ = 100x\]
Also, we know there are ‘y’ prizes of $125$ each
\[ \Rightarrow \]Total prize money for $25$ prizes\[ = 25y\]
Then we can form an equation using the sum of prize money equal to sum of prize money from each amount of prizes
\[ \Rightarrow 100x + 25y = 2700\]................… (2)
From equation (1) we can write \[y = 63 - x\] . Substitute this value of y in equation (2)
\[ \Rightarrow 100x + 25(63 - x) = 2700\]
Calculate the product in LHS of the equation
\[ \Rightarrow 100x + 1575 - 25x = 2700\]
Shift all constant values to RHS of the equation
\[ \Rightarrow 100x - 25x = 2700 - 1575\]
Calculate difference on both sides of the equation
\[ \Rightarrow 75x = 1125\]
Divide both sides of the equation by 75
\[ \Rightarrow \dfrac{{75x}}{{75}} = \dfrac{{1125}}{{75}}\]
Cancel same factors from numerator and denominator on both sides of the equation
\[ \Rightarrow x = 15\]
Now we can substitute this value of x in equation \[y = 63 - x\] to find value of y
\[ \Rightarrow y = 63 - 15\]
\[ \Rightarrow y = 48\]

\[\therefore \]Number of $100$ prizes is 15 and number of $25$ prizes is 48.

Note: Many students make mistakes while applying substitution methods when solving for variables of two equations. Keep in mind we have to find the value of one variable in terms of another variable using any of the two given equations but then we substitute that value of variable in another equation (not in the equation from which we have derived that value).