Q1)
is the function representing the weekly paycheck of salesman A a proportional relationship?
p - function
s - number of sales
Salesman A earns £65 per sale. Therefore if he earns s number of sales, the total amount he earns is 65*s = 65s
Lets assume for s number of sales he receives his maximum weekly commission, therefore the equation for A's weekly payment is;
p = 65s +1300
to determine if p and s are proportional to each other, it should take the form of equation p=ks
where k is a constant. If we graph p=ks, then k is the slope that is a constant and the line should pass through the origin, which means that y intercept is 0.
however in this equation there is an intercept of +1300, therefore p and s are not proportional values.
Q2)
For salesman B he has a weekly base salary of £300 and commission of £40 per sale. Then the salary is fixed however for s number of sales, the commission he will receive is 40*s = 40s
the function for B's payment is as follows;
p = 40s + 300
This equation if we take it as a graph equation with p and s as y and x coordinates respectively, the equation we get is of one with an intercept of +300
therefore if we graph it, the line will not pass through the origin which means that the p and s values are not proportional to each other.
Q3)
Salesman C receives a weekly salary of £900. This is a fixed amount and does not vary depending on the sales. Therefore this is anyway not proportional as p function is not affected by s value.
therefore p and s are not related, hence they aren't proportional.
p = 900
I suppose you just have to simplify this expression.
(2ˣ⁺² - 2ˣ⁺³) / (2ˣ⁺¹ - 2ˣ⁺²)
Divide through every term by the lowest power of 2, which would be <em>x</em> + 1 :
… = (2ˣ⁺²/2ˣ⁺¹ - 2ˣ⁺³/2ˣ⁺¹) / (2ˣ⁺¹/2ˣ⁺¹ - 2ˣ⁺²/2ˣ⁺¹)
Recall that <em>n</em>ª / <em>n</em>ᵇ = <em>n</em>ª⁻ᵇ, so that we have
… = (2⁽ˣ⁺²⁾ ⁻ ⁽ˣ⁺¹⁾ - 2⁽ˣ⁺³⁾ ⁻ ⁽ˣ⁺¹⁾) / (2⁽ˣ⁺¹⁾ ⁻ ⁽ˣ⁺¹⁾ - 2⁽ˣ⁺²⁾ ⁻ ⁽ˣ⁺¹⁾)
… = (2¹ - 2²) / (2⁰ - 2¹)
… = (2 - 4) / (1 - 2)
… = (-2) / (-1)
… = 2
Another way to get the same result: rewrite every term as a multiple of <em>y</em> = 2ˣ :
… = (2²×2ˣ - 2³×2ˣ) / (2×2ˣ - 2²×2ˣ)
… = (4×2ˣ - 8×2ˣ) / (2×2ˣ - 4×2ˣ)
… = (4<em>y</em> - 8<em>y</em>) / (2<em>y</em> - 4<em>y</em>)
… = (-4<em>y</em>) / (-2<em>y</em>)
… = 2
