Answer to Question #154433 in Financial Math for kris

Question

A retailer received an invoice for the purchase of twenty rackets at RM140 each. He was given trade discounts of 15% and 5%. The cash discount terms on the invoice were 3/10, n/30. (a) Determine the single trade discount rate that is equivalent to the chain discounts. (b) Calculate the amount of payment if it is made one week after the date of invoice.

Answer 1

$\bold {Answers}$

a) The single equivalent discount, $d = 19.25\%$

b) Amount of payment $= RM \space 2,193.17$

$\bold {Solutions}$

a) Let the chain discounts be $x\%$ and $y\%$ , and the single equivalent discount

be $d\%.$ Suppose the gross invoice price is $\$N,$ then, net invoice price: price after discounts will be given by the formula:

Net price $= \$N(1-\dfrac{x}{100})(1-\dfrac {y}{100})$

$(1-\dfrac{x}{100})(1-\dfrac {y}{100})$ is the factor after deducting discounts of x% and y%.

$Let \space (1-\dfrac{x}{100})(1-\dfrac {y}{100}) = z\%$

$=>$ $1 - \dfrac { z }{100}=$ total discount subtracted.

Expanding brackets on LHS gives:

$1 - \dfrac {x}{100} - \dfrac {y}{100} + \dfrac {xy}{10,000} = \dfrac {z}{100}$

$=> 1 - \dfrac {z}{100} = \dfrac {x}{100} + \dfrac {y}{100} - \dfrac {xy}{10,000}$

Multiplying each term by 100 to simplify, gives:

$100-z = x + y - \dfrac {xy}{100}$

$=> (100 - z)\% = (x + y) - \dfrac {xy}{100}$

Since $(100-z)\%$ is the discount, it follows that $d = 100 - z$

$=> d = (x + y) - \dfrac {xy}{100}$

Now, it is given that $x = 15, and \space y = 5$

$\therefore \space d = (15 + 5) - \dfrac {15×5}{100}$

$= 20\% - (15 × 0.05)\%$

$= 20\% - 0.75\%$

$=\bold { 19.25\%}$

$\bold {Alternatively}$

Let $\$N$ be the gross invoice price. Net invoice price $= \$N×(1-0.15)(1-0.05)$

$= \$N× 0.85×0.95$

$=\$0.8075N$

Therefore,

$Discount = 1-0.8075$

$= 0.1925 × 100\%$

$=\bold{ 19.25\%}$

b) Cash discount terms 3/10, n/30 imply that the retailer is entitled to a 3% cash discount is the debt is settled within 10 days from the date of the invoice, and the invoice should be settled not latter than 30 days. Since the amount was settled one week after the date of the invoice, the retailer satisfies the condition for a 3% cash discount.

Total invoice price = 20 rackets × RM140/racket

= RM 2,800

Therefore, amount paid is net of 19.25% trade discount and a 3% cash discount.

Amount paid $= RM \space 2,800 × (100\% - 19.25\%) × (100\% - 3\%)$

$= RM \space 2,800 × 80.75\% × 97\%$

$= RM \space 2,800 × 0.8075 × 0.97$

$= RM \space 2,800 × 0.783275$

$=\bold { RM \space 2,193.17}$