Marked price under successive discounts, sales tax inside a total charge, the area and perimeter of a rectangular campsite, and arranging fractions in descending order

Understanding the topic

Question 7 turns on one idea that examiners repeatedly catch candidates on: successive discounts multiply, they do not add. The geometry part is a Pythagoras equation in disguise, and the fractions part is fastest by cross-multiplication.

(a) Calculate the marked price of the sculpture if it is sold for Rs. 79,950/- after a discount of 18%. On another day, if 10% discount is given on the marked price of the same sculpture before it is sold at a further discount of 8%, would the sale price still be Rs. 79,950/-?

Given: an 18% discount produced a selling price of Rs. 79,950; on another day the discounts offered were 10% followed by a further 8%.

Method: after an 18% discount the selling price is 82% of the marked price, so divide to recover the marked price; then apply the two later discounts one after the other as multipliers.

Working:

Marked price = 79,950 ÷ 0.82 = 97,500

After the 10% discount: 97,500 × 0.90 = 87,750

After the further 8% discount: 87,750 × 0.92 = 80,730

Answer: the marked price was Rs. 97,500, and the second sale price would be Rs. 80,730, not Rs. 79,950.

It would be Rs. 780 higher. The combined successive discount is 1 - (0.90 × 0.92) = 17.2%, not 18%.

(b) Majid pays a total of Rs. 51,520/- for his car insurance. The total is made up of a basic charge plus 15% sales tax. Calculate the amount of sales tax that Majid pays.

Given: Majid paid Rs. 51,520 in total, which is the basic charge plus 15% sales tax.

Method: the total is 115% of the basic charge, so divide by 1.15 to recover the basic charge, then subtract.

Working:

Basic charge = 51,520 ÷ 1.15 = 44,800

Sales tax = 51,520 - 44,800 = 6,720

Answer: Majid pays Rs. 6,720 as sales tax.

Check: 15% of 44,800 = 6,720.

(c) A camp site in the shape of a rectangle has sides (3x + 6) m and (x + 1) m and the length of the diagonal is (4x + 1) m. Find the area and the perimeter of the camp site.

Given: the sides are (3x + 6) m and (x + 1) m, and the diagonal is (4x + 1) m.

Method: the diagonal of a rectangle is the hypotenuse of a right triangle formed by the two sides, so apply Pythagoras and solve the resulting quadratic in x. Note that x here is the unknown length variable, never a multiplication sign.

Working:

(3x + 6)² + (x + 1)² = (4x + 1)²

9x² + 36x + 36 + x² + 2x + 1 = 16x² + 8x + 1

x² - 5x - 6 = 0

(x - 6)(x + 1) = 0

x = 6; x = -1 is rejected because it gives an invalid side.

Dimensions: 3(6) + 6 = 24 m and 6 + 1 = 7 m

Area = 24 × 7 = 168 m²

Perimeter = 2(24 + 7) = 62 m

Answer: area = 168 m²; perimeter = 62 m.

Check: the diagonal is 4(6) + 1 = 25 m, and 7² + 24² = 49 + 576 = 625 = 25².

(d) Arrange the following fractions in descending order:

Given, as printed:

1. Set (a): 7/12, 4/7, 9/14, 25/42
2. Set (b): 5/8, 7/12, 5/9, 13/24
3. Set (c): 6/35, 5/21, 2/7, 4/15
4. Set (d): 2/5, 6/11, 7/15, 9/20, 13/25
5. Set (e): 1/5, 2/7, 3/8, 4/13, 5/17

Method: compare two fractions by cross-multiplying across positive denominators; the larger cross-product marks the larger fraction. For example, 9/14 > 25/42 because 9 × 42 = 378 while 25 × 14 = 350.

Answers:

1. Set (a): 9/14 > 25/42 > 7/12 > 4/7
2. Set (b): 5/8 > 7/12 > 5/9 > 13/24
3. Set (c): 2/7 > 4/15 > 5/21 > 6/35
4. Set (d): 6/11 > 13/25 > 7/15 > 9/20 > 2/5
5. Set (e): 3/8 > 4/13 > 5/17 > 2/7 > 1/5

A common denominator gives the same orders and is worth using as a check when two fractions are close.