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Formulas

5. Formulas. Case Study. 5.1 Sequences. 5.2 Introduction to Functions. 5.3 Simple Algebraic Fractions. 5.4 Formulas and Substitution. 5.5 Change of Subject. Chapter Summary. Case Study. Let us check your body mass index and see whether you are normal or not. Is my weight normal?.

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Formulas

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  1. 5 Formulas Case Study 5.1 Sequences 5.2 Introduction to Functions 5.3 Simple Algebraic Fractions 5.4 Formulas and Substitution 5.5 Change of Subject Chapter Summary

  2. Case Study Let us check your body mass index and see whether you are normal or not. Is my weight normal? Body Mass Index (BMI) is frequently used to check whether a person’s body weight and body height are in an appropriate proportion. It can be calculated by using the following formula: For example, if a person’s weight and height are 50 kg and 1.6 m respectively, then the BMI  50  1.62  19.5. The following table shows the range of the BMI and the corresponding body condition.

  3. 5.1 Sequences A. Introduction to Sequences John recorded the weights of 10 classmates. He listed their weights (in kg) in the order of their class numbers. We call such a list of numbers a sequence. Each number in a sequence is called a term. We usually denote the first term as T1, the second term as T2, and so on. In the above sequence, T1 46, T2  42, T3 50, ... , etc.

  4. +4 +4 2 2 5.1 Sequences B. Some Common Sequences Some sequences have certain patterns, but some do not. 1. Consider the sequence 1, 5, 9, 13, ... . We can guess the subsequent terms: 17, 21, 25, ... When the difference between any 2 consecutive terms is a constant, such a sequence is called an arithmetic sequence, and the difference is called a common difference. 2. Consider the sequence 8, 16, 32, 64, ... . We can guess the subsequent terms: 128, 256, 512, ... When the ratio of each term (except the first term) to the preceding term is a constant, such a sequence is called a geometric sequence, and the ratio is called a common ratio.

  5. 5.1 Sequences B. Some Common Sequences 3. We can arrange some dots to form some squares. The number of dots used in each square is called a square number. 4. We can arrange some dots to form some triangles. The number of dots used in each triangle is called a triangular number. 5. Consider the sequence 1, 1, 2, 3, 5, 8, ... . In this sequence, starting from the third term, each term is the sum of the 2 preceding terms. This sequence is called the Fibonacci sequence.

  6. It is a common practice towritethegeneralterm of a sequence as an algebraic expression in terms of n. 5.1 Sequences C. General Terms For a sequence that shows a certain pattern, we can use Tn to represent the nth term. Tn is called the general term of the sequence. For example, since the sequence 2, 4, 6, 8, 10, ... has consecutive even numbers, we can deduce that the general term of this sequence is Tn 2n. Once the general term of a sequence is obtained, we can use it to describe any term in a sequence.

  7. Find the general terms of the following sequences. (a) 7, 14, 21, 28, ... (b) 1, 2, 4, ... (c) (d) 15, 14, 13, 12, ... 5.1 Sequences C. General Terms Example 5.1T Solution: (a) The sequence 7, 14, 21, 28, ... can be written as 7(1), 7(2), 7(3), 7(4), ... ∴ The general term of the sequence is 7n. (b) The sequence 1, 2, 4, ... can be written as 21  1, 22  1, 23 1 ... ∴ The general term of the sequence is 2n  1.

  8. Find the general terms of the following sequences. (a) 7, 14, 21, 28, ... (b) 1, 2, 4, ... (c) (d) 15, 14, 13, 12, ... (c) The sequence can be written as ∴ The general term of the sequence is . 5.1 Sequences C. General Terms Example 5.1T Solution: (d) The sequence 15, 14, 13, 12, ... can be written as 16  1, 16  2, 16  3, 16  4, ... ∴ The general term of the sequence is 16 n.

  9. +3 +3 +3 Rewrite the terms of the sequence as expressions in which the order of the terms can be observed. T9 3(9)  4  23 T9 92  81 5.1 Sequences C. General Terms Example 5.2T Find thegeneraltermand the9thtermofeachof thefollowingsequences. (a) 1, 2, 5, 8, ... (b) 1, 4, 9, 16, ... Solution: (a) T1 1 T2 2 T3 5 T4  8  3  4  6  4  9  4  12  4  3(1)  4  3(2)  4  3(3)  4  3(4)  4 (b) T1  1 T2 4 T3 9 T4  16  12  22  32  42 ∴ Tn 3n 4 ∴ Tnn2

  10. 5.2 Introduction to Functions For each input value of n, there is exactly one output value of Tn. In the previous section, we learnt how to find the values of the terms in a sequence from the general term by substituting different values of n in the general term. Consider a sequence with the general term Tn 5n 2. The above figure shows an ‘input-process-output’ relationship, which is called a function. In this example, we call Tn a function of n.

  11. 5.2 Introduction to Functions The idea of function is common in our daily lives. Suppose that each can of cola costs $5. Let x be the number of cans of cola, and $y be the corresponding total cost. Since the total cost of x cans of cola is $5x, the equation y 5x represents the relationship between x and y. The following table shows some values of x and the corresponding values of y. For every value of x, there is only one corresponding value of y. We say that y is a function of x.

  12. If p is a function of q such that p 4q 5, find the values of p for the following values of q. (a) 6 (b) 5 (c)  19 Substitute q  6 into the expression 4q  5. 25 (c) When q , p    5.2 Introduction to Functions Example 5.3T Solution: (a) When q 6, p  4(6)  5 (b) When q5, p  4(5)  5

  13. We learnt at primary level that numbers in the form , where a and b are integers and b 0, are called fractions. Note that is not an algebraic fraction because the denominator is a constant. 5.3 Simple Algebraic Fractions A. Simplification When both the numerator and the denominator of a fractional expression are polynomials, where the denominator is not a constant, such as: we call such an expression an algebraic fraction.

  14. 5.3 Simple Algebraic Fractions A. Simplification We can simplify a numerical fraction, whose numerator and denominator both have common factors, by cancelling the common factors. For example: For algebraic fractions, we can simplify them in a similar way, when the common factors are numbers, variables or polynomials. For example:

  15. Simplify the following algebraic fractions. (a) (b) First factorize the numerator and the denominator. Then cancel out the common factors. We cannot cancel common factors from the terms 7y and 21y only, i.e., the fraction cannot be simplified as . 3 5.3 Simple Algebraic Fractions A. Simplification Example 5.4T Solution:

  16. Simplify the following algebraic fractions. (a) (b) First factorize the numerator. You may check if 2k + h (the denominator) is a factor of the numerator. 5.3 Simple Algebraic Fractions A. Simplification Example 5.5T Solution:

  17. 5.3 Simple Algebraic Fractions B. Multiplication and Division When multiplying or dividing a fraction, we usually try to cancel out all common factors before multiplying the numerator and the denominator separately to get the final result. For example: We can perform the multiplication or division of algebraic fractions in a similar way. For example:

  18. Simplify the following algebraic fractions. (a) (b) 5.3 Simple Algebraic Fractions B. Multiplication and Division Example 5.6T Solution:

  19. 5.3 Simple Algebraic Fractions C. Addition and Subtraction The method used for the addition and subtraction of algebraic fractions is similar to that for numerical fractions. For example: When the denominators of algebraic fractions are not equal, first we have to find the lowest common multiple (L.C.M.) of the denominators. For example:

  20. Simplify . 5.3 Simple Algebraic Fractions C. Addition and Subtraction Example 5.7T Solution:

  21. Simplify . For any number x 0, 5.3 Simple Algebraic Fractions C. Addition and Subtraction Example 5.8T Solution:

  22. Simplify . Since 3(p 3) and p have no common factors other than 1, the L.C.M. of 3(p 3) and pis 3p(p 3). 5.3 Simple Algebraic Fractions C. Addition and Subtraction Example 5.9T Solution:

  23. 5.4 Formulas and Substitution Consider the volume (V) of a cuboid: Vlwh where l is the length, w is the width and h is the height. If the values of the variables l, w and h are already known, we can find V by the method of substitution. Similarly, if the values of V, l and w are known, we can find h. Actually, in any given formula, we can find any one of the variables by the method of substitution if all the others are known.

  24. Consider the formula . Find the value of h if T 121 and r 2.5. Factorize the expression. 5.4 Formulas and Substitution Example 5.10T Solution:

  25. Given the area (A) of a triangle: where b is the base length and h is the height. In another case, if we need to find h when A and b are known, it is more convenient to use another formula , with h being the subject. The process of obtaining the formula from is called the change of subject. 5.5 Change of Subject In the formula, A is the only variable on the left-hand side. We call A the subject of the formula. This formula can be used if we want to find A, when b and h are known. We can use the method of solving equations to change the subject of a formula.

  26.  4 is transposed to the L.H.S. to become  4. Rewrite the formula such that only the variable u is on the L.H.S. First move the other terms to one side such that only the variable u remains on the other side. 5.5 Change of Subject Example 5.11T Make u the subject of the formula 3k 4  5u. Solution:

  27. Make m the subject of the formula . Simplify the fractions. The L.C.M. of r and p is pr. Multiply both sides by mpr. 5.5 Change of Subject Example 5.12T Solution:

  28. Make m the subject of the formula . Remove the brackets Take out the common factor m. Firstmovealltheterms that involve m to one side. 5.5 Change of Subject Example 5.13T Solution:

  29. A bag of food is put into a refrigerator. The temperature T (in C) of the food after time t (in hours) is given by the formula . (a) Make t the subject of the formula. (b) How long will it take for the temperature of the food to become –3C? (a) (b) 5.5 Change of Subject Example 5.14T Solution: ∴ It takes 6 hours for the temperature of the food to become 3C.

  30. Chapter Summary 5.1 Sequences A list of numbers arranged in an order is called a sequence. Each number in a sequence is called a term. For a sequence with a certain pattern, we can represent the sequence by its general term.

  31. 5.2 Introduction to Functions Chapter Summary A function describes an ‘input-process-output’ relationship between 2 variables. Each input gives only one output.

  32. 5.3 Simple Algebraic Fractions Chapter Summary The manipulations of algebraic fractions are similar to those of numerical fractions.

  33. 5.4 Formulas and Substitution Chapter Summary A formula is any equation that describes the relationship of 2 or more variables. By the method of substitution, we can find the value of a variable in a formula when the other variables are known.

  34. 5.5 Change of Subject Chapter Summary When there is only one variable on one side of a formula, this variable is called the subject of the formula.

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