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Solve the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs. 750. We would like to find out the number of toys produced on that day.
To do:
We have to solve the given situations mathematically.
Solution:
(i) John and Jivani together have $45$ marbles.
Both of them lost $5$ marbles each, and the product of the number of marbles they now have is $128$.
Let the number of marbles with John be $x$.
This implies,
Number of marbles with Jivani $=45-x$
Number of marbles John had after losing 5 marbles $= x - 5$
Number of marbles Jivani had after losing 5 marbles $= (45 - x) - 5 = 40 - x$
The product of the marbles they now have $=128$.
Therefore,
$(x - 5)(40 - x) = 128$
$40x-x^2-200+5x = 128$
$x^2 - 45x + 128 + 200 = 0$
$x^2 - 45x + 328 = 0$
$x^2-9x-36x+328=0$
$x(x-9)-36(x-9)=0$
$(x-9)(x-36)=0$
$x-9=0$ or $x-36=0$
$x=9$ or $x=36$
The number of marbles they had to start with is 9 and 36.
(ii) A cottage industry produces a certain number of toys in a day.
The cost of production of each toy (in rupees) was found to be $55$ minus the number of articles produced in a day.
On a particular day, the total cost of production was Rs. $750$.
Let the number of toys produced in a day be $x$.
This implies,
The cost of production of each toy $= 55 - x$
The total cost of production is the product of the number of toys produced in a day and the cost of production of each toy $ =x (55 - x)$
Therefore,
$x(55-x) = 750$
$55x-x^2 = 750$
$x^2-55x+750 = 0$
$x^2-25x-30x+750=0$
$x(x-25)-30(x-25)=0$
$(x-25)(x-30)=0$
$x-25=0$ or $x-30=0$
$x=25$ or $x=30$
The number of toys produced on that day was 25 or 30.