Review: proportion, percent, sets, integers and rational numbers
Mathematics Prep 1 Term 1, Units 1–4, Review
GoalRevise Unit 1 with new mixed tasks.
New words
proportion · two equal ratios, tested by equal cross productsunion and intersection · all the elements in either set, and the elements in bothintegers · the whole numbers with their opposites: …, −2, −1, 0, 1, 2, …rational numbers · numbers that can be written as a fraction a/b with b ≠ 0
Explanation
Proportion: a : b = c : d when a × d = b × c; cross multiply and divide to find a missing term, and keep the same order in both ratios. Scale is drawing length : real length in the same unit. To share an amount in a ratio, add the parts, find one part and multiply. A discount of d% means you pay (100 − d)% of the price; tax of 14% adds 14% of the price. Sets: A ∪ B has every element once and A ∩ B has the common ones, so n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Integers: same signs add and keep the sign, subtracting means adding the opposite, and the product of two numbers with the same sign is positive. Rational numbers: use a common denominator to add and the inverse of the second number to divide.
ƒFormula
a : b = c : d means a × d = b × c
Percent change: new = old × (100 ± p) ÷ 100.
Worked examples
Proportion. 12 notebooks cost 90 LE. The price x of 20 notebooks: 12 : 90 = 20 : x, so 12 × x = 90 × 20, x = 1800 ÷ 12 = 150 LE. Check: one notebook costs 90 ÷ 12 = 7.5 LE and 20 × 7.5 = 150.
Sharing and tax. Karim and Dina share 450 LE in the ratio 4 : 5. There are 4 + 5 = 9 parts and 450 ÷ 9 = 50 LE per part, so Karim gets 200 LE and Dina gets 250 LE. A price of 200 LE plus 14% tax: tax = 200 × 14 ÷ 100 = 28 LE and the total is 228 LE.
Original price. A jacket is sold for 255 LE after a 15% discount. The sale price is 85% of the original, so the original is 255 ÷ 85 × 100 = 3 × 100 = 300 LE. Check: 15% of 300 is 45 and 300 − 45 = 255.
Sets. A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}. A ∩ B = {2, 4, 6}. A ∪ B = {1, 2, 3, 4, 5, 6, 8}. Check the count: 6 + 4 − 3 = 7.
Receipt maths. With a grown-up, take a shopping receipt. Pick three items and find the total, the share of the total spent on each as a percent, and what the total would be after a 10% discount.
Practice
1
Is 6 : 15 = 10 : 25? Show your working.
6 × 25 = 150 and 15 × 10 = 150. The cross products are equal, so yes.
2
On a map with scale 1 : 50,000, two towns are 4.5 cm apart. How many kilometres is it in real life?
4.5 × 50,000 = 225,000 cm = 2,250 m = 2.25 km.
3
A = {x : x ∈ N, x < 5} and B = {3, 4, 5, 6}. List A, A ∪ B and A ∩ B.
A = {0, 1, 2, 3, 4} because N starts at 0. A ∪ B = {0, 1, 2, 3, 4, 5, 6}. A ∩ B = {3, 4}.
4
Calculate −7 − (−12) + 3 × (−2) and then 1/2 − 3/4 ÷ 3/8.
