# Question f5a58

Sep 19, 2017

8

#### Explanation:

Let friends be x and bill comes Rs. 1120. So 1 has to pay Rs. 1120/x.

As 3 have no wallets, so ( x - 3 ) friends have to pay Rs. 84 each as an extra. Means Rs (1120/x + 84) each.

Now as per question,
$\left(\frac{1120}{x} + 84\right) \left(x - 3\right) = 1120$

$\Rightarrow \frac{1120}{x} . x + 84 x - \frac{3.1120}{x} - 84.3 = 1120$

$\Rightarrow 84 x - \frac{3360}{x} = 1120 - 1120 + 252$

$\Rightarrow 84 {x}^{2} - 252 x - 3360 = 0$

$\Rightarrow 84 \left({x}^{2} - 3 x - 40\right) = 0$

$\Rightarrow {x}^{2} - 3 x - 40 = 0$

$\Rightarrow {x}^{2} - 8 x + 5 x - 40 = 0$

$\Rightarrow x \left(x - 8\right) + 5 \left(x - 8\right) = 0$

$\Rightarrow \left(x - 8\right) \left(x + 5\right) = 0$

$\Rightarrow x - 8 = 0 , x + 5 = 0$

x = 8 , -5 [ x cannot be -5]

hence x = 8

SO, TOTAL FRIENDS BE 8

Sep 19, 2017

Eight.

#### Explanation:

This situation can be expressed as the equation

$T = \left(\setminus \frac{T}{P} + E\right) \left(P - 3\right)$

where $T$ is the total dinner price, $P$ is the number of people and $E$ is the extra money.

(Think about this until it makes sense to you. The point of this problem is to practice converting a situation into an equation that models it.)

Each person who is paying is paying what they would have paid if everyone was paying, plus an extra $84#. That is, each person is paying $\setminus \frac{T}{P}$(what they would have paid if the price was evenly divided between the friends), plus some extra amount $E$, so $\setminus \frac{T}{P} + E$each in total. To find how much that is collectively, multiply by the number of people paying that much, $P - 3$. This should be equal to the total price for the dinner. From here, simply expand the equation and solve for $P$using the quadratic formula. $T = \left(\setminus \frac{T}{P} + E\right) \left(P - 3\right)$$T = T + E P - \setminus \frac{3 T}{P} - 3 E$$0 = E P - \setminus \frac{3 T}{P} - 3 E$$0 = E {P}^{2} - 3 E P - 3 T$Substitute $E = 84$and $T = 1120$and solve using the quadratic formula for finding roots of equations of the form $a {x}^{2} + b x + c$: $P = \setminus \frac{- b \setminus \pm \setminus \sqrt{{b}^{2} - 4 a c}}{2 b}$$P = \setminus \frac{3 E \setminus \pm \setminus \sqrt{9 {E}^{2} + 12 E T}}{2 E}$$P = \setminus \frac{1344}{168}$$P = 8\$