Properties of Integers, Fractions, Decimals and Percentages for IB Board Class 9

Properties of Integers with Examples

Integers are the set of whole numbers and their negatives, including zero. They follow important mathematical properties that ensure consistency in solving equations and algebraic problems. These include closure, existence of additive inverses, commutative and associative laws. Below, each property is explained with examples.

1. Closure Property

The set of integers is closed under addition, subtraction, and multiplication. This means performing these operations on any two integers always gives another integer.

Example 1: Addition

-7 + 12 = 5 (Result is also an integer)

Example 2: Multiplication

(-6) × 4 = -24 (Result is also an integer)

Note: Division is not always closed. For example, 7/2 is not an integer.

2. Additive Inverse Property

Every integer has an additive inverse, i.e., for any integer a, there exists -a such that:

a + (-a) = 0

Example 1: The additive inverse of 15 is -15, since 15 + (-15) = 0.

Example 2: The additive inverse of -9 is 9, since -9 + 9 = 0.

3. Commutative Laws

The order of numbers does not affect the result in addition and multiplication.

a + b = b + a, a × b = b × a

Example 1: Addition

(-8) + 13 = 5, 13 + (-8) = 5

Example 2: Multiplication

(-7) × 6 = -42, 6 × (-7) = -42

4. Associative Laws

The grouping of numbers does not affect the result in addition and multiplication.

(a+b)+c = a+(b+c), (a × b)×c = a × (b × c)

Example 1: Addition

((-3) + 5) + 7 = 2 + 7 = 9
(-3) + (5 + 7) = -3 + 12 = 9

Example 2: Multiplication

((-2) × 4) × 3 = (-8) × 3 = -24
(-2) × (4 × 3) = (-2) × 12 = -24

Both groupings give the same result.

Solved Examples:

Example 1: Simplify (-12) + (+25) – (-8) step by step.

(-12) + (+25) – (-8) = -12 + 25 + 8 = 21

Example 2: Verify whether multiplication of integers is commutative using (-6) × 4 and 4 × (-6).

(-6) × 4 = -24 and 4 × (-6) = -24

Since the results are equal, multiplication is commutative.

Fractions and Their Properties

A fraction represents a part of a whole and is written as a/b, where a is the numerator and b ≠ 0 is the denominator. Fractions can be simplified, compared, or expressed as equivalent fractions. They are used in daily life for dividing quantities, measuring, and in higher mathematics such as probability and algebra. The four fundamental operations on fractions are: simplification, addition/subtraction, multiplication, and division. Below are the properties with step-by-step examples.

1. Simplification of Fractions

Fractions can be reduced to simplest form by dividing numerator and denominator by their greatest common divisor (GCD).

Example: Simplify 84/126.

84/126 = (84 ÷ 42)/(126 ÷ 42) = 2/3

2. Equivalent Fractions

Two fractions are equivalent if their cross-products are equal, i.e., a/b = c/d when a × d = b × c.

Example: Check whether 3/4 and 9/12 are equivalent.

3 × 12 = 36, 4 × 9 = 36

Since both products are equal, the fractions are equivalent.

3. Addition and Subtraction of Fractions

Fractions must have a common denominator before they can be added or subtracted.

Example: Add 5/6 and 7/9.

5/6 + 7/9 = 15/18 + 14/18 = 29/18 = 1 11/18

Example: Subtract 7/8 – 5/12.

7/8 – 5/12 = 21/24 – 10/24 = 11/24

4. Multiplication of Fractions

To multiply fractions, multiply numerators together and denominators together.

Example: Multiply 5/9 × 27/40.

5/9 × 27/40 = 135/360 = 3/8

5. Division of Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second.

Example: Divide 15/16 ÷ 5/8.

15/16 ÷ 5/8 = 15/16 × 8/5 = 120/80 = 3/2

Solved Examples:

Example 1: Simplify 84/126.

84/126 = (84 ÷ 42)/(126 ÷ 42) = 2/3

Example 2: Add 5/6 and 7/9.

5/6 + 7/9 = 15/18 + 14/18 = 29/18 = 1 11/18

Decimals and Their Operations

Decimals are numbers expressed in the base-10 system using a decimal point. They represent tenths, hundredths, thousandths, and so on. Decimals can be of two types:

  • Terminating decimals: These have a finite number of digits after the decimal point, e.g., 0.75.
  • Recurring decimals: These have digits repeating infinitely, e.g., 0.3̅.

Decimals are widely used in real life for money, measurements, and scientific data. The four main operations on decimals are addition, subtraction, multiplication, and division.

1. Converting Fractions to Decimals

Fractions can be expressed as decimals by dividing the numerator by the denominator.

Example: Convert 7/8 into decimal.

7 ÷ 8 = 0.875

Hence, 7/8 = 0.875.

2. Converting Decimals to Fractions

Decimals can be written as fractions in simplest form.

Example: Express 0.125 as a fraction.

0.125 = 125/1000 = (125 ÷ 125)/(1000 ÷ 125) = 1/8

3. Addition and Subtraction of Decimals

Align the decimal points and then perform the operation.

Example: Add 12.45 and 8.7.

12.45 + 8.70 = 21.15

Example: Subtract 35.46 – 18.729.

35.460 – 18.729 = 16.731

4. Multiplication of Decimals

Multiply the numbers ignoring the decimal points. Then place the decimal in the product so that the total number of decimal places equals the sum of decimal places in the factors.

Example: Multiply 12.6 × 4.8.

126 × 48 = 6048

Since 12.6 has 1 decimal place and 4.8 has 1 decimal place, the product must have 2 decimal places.

12.6 × 4.8 = 60.48

5. Division of Decimals

Shift the decimal point in both numbers to make the divisor a whole number, then divide as usual.

Example: Divide 45.6 ÷ 1.2.

45.6/1.2 = 456/12 = 38

Solved Examples:

Example 1: Convert 13/40 into decimal.

13/40 = (13 × 25)/(40 × 25) = 325/1000 = 0.325

Example 2: Multiply 12.6 × 4.8.

12.6 × 4.8 = 126 × 48 ÷ 100 = 6048 ÷ 100 = 60.48

Percentages

A percentage expresses a number as a fraction of 100. It is useful for comparisons, profit and loss, discounts, and exam scores. The formula is:

Percentage = (Part/Whole) × 100

Solved Examples:

Example 1: A school has 450 students. If 180 are girls, what percent are girls?

(180/450) × 100 = 40%

Example 2: The price of a bicycle was $1200 and increased to $1380. Find the percentage increase.

Increase = 1380 – 1200 = 180
Percentage increase = (180/1200) × 100 = 15%

Grade 9 – IB Board

Chapter: Properties of Integers, Fractions, Decimals and Percentages

Level 1

  1. A submarine is at a depth of $-325$ m below sea level. It rises by $178$ m and then dives $97$ m deeper. What is its final position relative to sea level?
    Answer: $-244$ m
  2. A shopkeeper had a loss of \$185 one day and a profit of \$276 the next day. What was his overall balance after the two days?
    Answer: \$91
  3. The temperature at a hill station was $-12^\circ$C in the morning. By noon it increased by $15^\circ$C, and by night it dropped by $19^\circ$C. What was the temperature at night?
    Answer: $-16^\circ$C
  4. A company made a profit of \$8,400 in January and a loss of \$12,600 in February. What is the overall result for the two months?
    Answer: Loss of \$4,200
  5. A family ate $\dfrac{3}{8}$ of a cake in the morning and $\dfrac{1}{4}$ in the evening. What fraction of the cake was eaten in total?
    Answer: $\dfrac{5}{8}$
  6. A bottle contains $\dfrac{7}{10}$ litre of juice. If a person drinks $\dfrac{2}{5}$ litre from it, how much juice remains in the bottle?
    Answer: $\dfrac{3}{10}$ litre
  7. Out of a class of 60 students, $\dfrac{2}{5}$ are girls. How many boys are there in the class?
    Answer: 36
  8. A student scored $18$ marks out of $30$ in Mathematics, $42$ marks out of $50$ in Science and $70$ marks out of $100$ in English. Find his overall percentage.
    Answer: 72.22%
  9. A shopkeeper increases the price of a shirt by $20\%$. If the original price was \$750, find the new price.
    Answer: \$900
  10. A book is marked at \$560 and is sold at a discount of $15\%$. Find the selling price of the book.
    Answer: \$476
  11. A laptop is first increased in price by $10\%$ and then decreased by $10\%$. Find the net percentage change in its price.
    Answer: 1% decrease
  12. A student got 30 questions correct out of 40 in a test. What percentage of questions did he answer correctly?
    Answer: 75%
  13. A person saves $\dfrac{3}{7}$ of his monthly income. If his monthly income is \$21,000, find the amount he saves.
    Answer: \$9,000
  14. The population of a village was 24,000. It increased by $12.5\%$ in a year. Find the population at the end of the year.
    Answer: 27,000
  15. A man buys an article for \$1,500 and sells it at a loss of $12\%$. Find the selling price of the article.
    Answer: \$1,320

Solution 1: Depth $= -325 + 178 – 97 = -244$ m. Answer matches.

Solution 2: $-185 + 276 = 91$. Answer matches.

Solution 3: $-12 + 15 – 19 = -16^\circ$C. Answer matches.

Solution 4: $8400 – 12600 = -4200$. Overall result is a loss of \$4200. Answer matches.

Solution 5: $\dfrac{3}{8} + \dfrac{1}{4} = \dfrac{3}{8} + \dfrac{2}{8} = \dfrac{5}{8}$. Answer matches.

Solution 6: $\dfrac{7}{10} – \dfrac{2}{5} = \dfrac{7}{10} – \dfrac{4}{10} = \dfrac{3}{10}$. Answer matches.

Solution 7: Number of girls $= \dfrac{2}{5} \times 60 = 24$. Boys $= 60 – 24 = 36$. Answer matches.

Solution 8: Total marks obtained $= 18 + 42 + 70 = 130$. Total marks $= 30 + 50 + 100 = 180$. Percentage $= \dfrac{130}{180} \times 100 = 72.22\%$. Answer matches.

Solution 9: $750 + 20\% \times 750 = 750 + 150 = 900$. Answer matches.

Solution 10: Discount $= 15\% \times 560 = 84$. Selling price $= 560 – 84 = 476$. Answer matches.

Solution 11: Price after increase $= 100 \to 110$. Decrease $10\%$ of 110: $110 \to 99$. Net decrease $= 1\%$. Answer matches.

Solution 12: $\dfrac{30}{40} \times 100 = 75\%$. Answer matches.

Grade 9 – IB Board

Chapter: Properties of Integers, Fractions, Decimals and Percentages

Level 2

  1. A submarine dives to $-450$ m, then rises by $230$ m, and later descends another $175$ m. What is its final depth?
    Answer: $-395$ m
  2. In a football tournament, Team A gained $+35$ points in one match, lost $-28$ points in another, and gained $+16$ points in the final match. What is the team’s net score?
    Answer: 23 points
  3. A scientist recorded the pH of a solution as $-2$. After adding a chemical, the value increased by 5. On further heating, it decreased by 7. What is the final pH value?
    Answer: $-4$
  4. A mountain climber was at an altitude of 2450 m. He descended 875 m, climbed 560 m, and again descended 940 m. What is his final altitude?
    Answer: 1195 m
  5. A pizza was cut into 12 equal slices. If Riya ate $\tfrac{1}{3}$ of the pizza and her friend ate $\tfrac{1}{4}$ of the pizza, how many slices are left?
    Answer: 5 slices
  6. A recipe requires $\tfrac{3}{5}$ cup of flour, $\tfrac{1}{4}$ cup of sugar, and $\tfrac{2}{5}$ cup of milk. How much ingredient is used in total?
    Answer: $\tfrac{23}{20}$ cups
  7. A swimming pool was filled to $\tfrac{7}{8}$ of its capacity. After some water was drained, it remained $\tfrac{5}{12}$ full. What fraction of the pool’s capacity was emptied?
    Answer: $\tfrac{11}{24}$
  8. A marathon runner covered 12.75 km in the first hour, 14.5 km in the second, and 11.25 km in the third. How much distance did he run in total?
    Answer: 38.5 km
  9. A tailor bought 37.5 m of cloth. He used 12.8 m for shirts, 9.75 m for trousers, and 8.5 m for coats. How much cloth is still left?
    Answer: 6.45 m
  10. A library has 18,000 books. If 35% are science books and 40% are literature books, how many books belong to other categories?
    Answer: 4,500 books
  11. A farmer spent 22% of his income on seeds, 18% on fertilizers, and 25% on equipment. If his income was \$80,000, how much money did he save?
    Answer: \$28,000
  12. A student scored 78% in Mathematics, 64% in English, and 83% in Science, each subject out of 100 marks. Find his average percentage score.
    Answer: 75%
  13. The population of a city was 45,000. It first increased by 20% in one year and then decreased by 10% the following year. Find the population at the end of the two years.
    Answer: 48,600
  14. A trader bought an article for \$6,000. He marked it at \$8,000 and gave a 12.5% discount. Find his percentage profit.
    Answer: 25%
  15. A car’s value depreciates by 15% each year. If the present value is \$340,000, what will be its value after 2 years?
    Answer: \$244,900

Solution 1: $-450 + 230 – 175 = -395$. Answer matches.

Solution 2: $35 – 28 + 16 = 23$. Answer matches.

Solution 3: $-2 + 5 – 7 = -4$. Answer matches.

Solution 4: $2450 – 875 + 560 – 940 = 1195$. Answer matches.

Solution 5: $\tfrac{1}{3} \times 12 = 4$, $\tfrac{1}{4} \times 12 = 3$, left $= 12 – 7 = 5$. Answer matches.

Solution 6: $\tfrac{3}{5} + \tfrac{1}{4} + \tfrac{2}{5} = \tfrac{23}{20}$. Answer matches.

Solution 7: $\tfrac{7}{8} – \tfrac{5}{12} = \tfrac{11}{24}$. Answer matches.

Solution 8: $12.75 + 14.5 + 11.25 = 38.5$ km. Answer matches.

Solution 9: $37.5 – (12.8 + 9.75 + 8.5) = 6.45$ m. Answer matches.

Solution 10: $18,000 \times (1 – 0.35 – 0.40) = 4,500$ books. Answer matches.

Solution 11: Total expenditure = 22% + 18% + 25% = 65%, savings = 35% of 80,000 = 28,000. Answer matches.

Solution 12: Average percentage = (78+64+83)/3 = 75%. Answer matches.

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