Writing Formulas and Substituting in a Formula

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8.1Writing Formulas and Substituting in a Formula

In this Math tutorial, you will learn:

  • What is a mathematical sentence?
  • What are mathematical operators?
  • Where does a mathematical sentence differ from a wordy sentence?
  • What are formulas? Why do we use them?
  • How to write a formula? What do we call its components?
  • What types of variables are there in a formula?
  • How to substitute the variables in a formula?
  • What is the procedure used when solving an exercise containing formulas?
  • How to convert wordy problems into formulas and vice-versa?
  • How to find the inverse formula when the original formula is known?

Introduction

How do you find the area of a square? Rectangle? Circle? Do you express the entire sentence in words or would you use a shorter method for this?

How do you express the fact that speed in a uniform motion is calculated by dividing distance and time?

How do you calculate the price of a product if the cost and profit are both known?

What do you understand by the term 'formula'? Could you remember two formulas from physics and two from chemistry? What about finance and banking? Are there any formulas in these fields?

This tutorial is focused on writing formulas and substituting quantities in a formula to find an unknown quantity in an easy way. Giving that formulas are present in many applied sciences, this is a very important topic with many applications in daily life.

Mathematical vs Word Sentences

A mathematical sentence is a fact (it may be either true or false) that combines two expressions (written in mathematical symbols) connected through a comparison operator between them. This comparison operator may be one of the following:

Equal to (=);

Greater than (>);

Smaller than (<);

Greater than or equal to (); and

Smaller than or equal to ()

Mathematical sentences are used to express word sentences in a much shorter way, where words are replaced with math symbols. Obviously, the symbols used in formulas must be widely recognized by scientists; that is, a specific math symbols conform to an agreed global standard and are used to express the same quantity in all textbooks. Thus, instead of writing, "Speed is equal to distance travelled in a given time", we simply write "v = d/t" where v is the symbol used for speed, d shows the travelling distance and t represents the time taken for the travel.

What is a Formula? How do we Write Formulas?

A formula is a fact or a rule written in mathematical symbols. It contains a set of instructions on how to calculate an unknown quantity in terms of one or more known ones. The unknown quantity (otherwise known as the independent variable) is related to the known ones (dependent variables) through the equal sign '='. Therefore, in a certain sense, a formula is a kind of equation (we will explain this concept in the next chapter) where variables are not just mathematical symbols but they have a more concrete and practical meaning. Hence, a formula is a kind of scientific "recipe" where the known variables are like the ingredients, which when combined give the final product (the unknown quantity).

As we said earlier, variables in formulas are combined with each other through mathematical operators such as addition, subtraction, multiplication, division, raise in power, logarithm, exponentiation, etc. Only the multiplication symbol ('·' or '×') is not written in a formula but is implied. Likewise, the division symbol (÷) is replaced by the fraction bar ('' or '/').

Example 1

Write down the following mathematical sentences as formulas using the given rules.

  1. Volume V of a cylinder is the product of base area Ab and height H, where the base area is a circle of radius R, and is obtained by raising the base radius to the second power and multiplying it by the Archimedes Constant π. You can use the Cylinder Calculator to see full calculations and formula in practice.
  2. Acceleration a of a moving object is the change in velocity in a given time t, where v0 is the initial velocity and v the final velocity of the moving object. You can use the Uniformly Accelerated Motion Calculator to see full calculations and formula in practice.

Solution 1

  1. From the description given in words, we obtain the following formula of cylinder volume:
    V = Ab∙H = πR2 H
    (You can see that multiplication symbol is obsolete, as explained earlier.)
  2. Since the term "change" means "difference" and it implies subtraction of the initial quantity from the final one, we obtain the following formula for acceleration:
    a = v - v0/t

We can also complete the inverse to interpret a formula and express the phenomenon described by it in words. Let's consider the example below to clarify this point.

Example 2

Describe in words the meaning of the following formulas.

  1. Formula of percentage of a substance A in a compound:
    %A = m(A)/m(tot) × 100%
    where m stands for 'mass' of substance.
  2. Formula of the heat energy Q dissipated by an electric device connected to a circuit operating at the current I:
    Q = I2 Rt
    where R is the resistance of the device and t is the operating time.

Solution 1

  1. The first formula indicates the ratio of the substance A to the total amount of the compound expressed as a percentage. In words, it means, "The percentage of a given substance to the total amount in a compound expressed as a percentage is obtained by dividing the mass of the given substance to the total mass of the compound and then multiplying the result by 100."
  2. This formula indicates that the heat energy dissipated by an electric device is the product of the square of operating current in the circuit, resistance of the device and the operating time.

Expressing Events in a Shorter Way using Formulas

We don't just use formulas to express scientific phenomena, they are also used to describe daily activities in a shorter way, especially when these activities are recurring and have the same routine. You may have used spreadsheets (like Microsoft Excel) to express a number of events involving the same variables in different situations. Thus, after writing the formula, you simply insert the input values and as a result, the output values appear automatically in the in the designated cells as shown in the figure below, where the average of all input values is calculated through the corresponding formula.

Math Tutorials: Writing Formulas and Substituting in a Formula Example

Thus, instead of adding all values manually and dividing them by the number of values to calculate the average, we simply insert the input values in different cells (here from B3 to B12) and by means of the formula of average shown in the top-right part of the figure, we calculate the average of the given numbers (here 59, shown in the cell B14) with just a click.

Indeed, if we add all numbers manually, the sum is 590. Since there are 10 numbers in total, the average is 59 (590 ÷ 10). However, the formula method is more suitable as we simply replace all input values with others to obtain the average without having to add them manually.

Likewise, many companies use formulas to calculate the cost of services they provide to customers. This saves a lot of precious time for employees who deal with customers. For example, if the cost of a drink in a restaurant table is D = $2.5 and the place reservation per person costs R = $6, the bill B is calculated by the formula

B = N∙ (R + n ∙ D)

or more specifically,

B = N ∙ (6 + n ∙ 2.5)

where n is the number of drinks a person orders. In this scenario, we have assumed that all customers have ordered the same number of drinks. In this way, the waiter has to insert only the number of customers N in the restaurant table and the number of drinks per person n to calculate the bill.

Example 3

Matt has £4,700 savings in the bank. He gets a job and is paid £12 for every hour he works. Assuming he spends nothing, write a formula for the amount of money (£M) Matt will have after he has worked for h hours.

Solution 3

The amount of money Matt will have after working for h hours in the new job will add to his savings to give the total amount he will have. Thus, we can write

Total amount = Initial amount + Hourly salary × Hours worked

When written in symbols, this relation becomes

(£M) = £4,700 + £12 × h

On the other hand, when written as a formula, the above relation becomes

M = 4,700 + 12h

where M is the total amount of money Matt will have after h hours.

Substituting into a Formula

The next step after writing (correctly) a formula is to substitute the known variables in it. In this way, we open the path to the calculation of the unknown quantity. For example, in the previous exercise we can substitute h = 20 and calculate how much money Matt will have in total after 20 hours of work. Thus, since the general rule applied in this situation corresponds to the formula we wrote at the end of the solved example,

M(h) = 4,700 + 12h

where M(h) represents the amount of money Matt will have after working for h hours, substituting h = 20 yields

M(20) = £4,700 + £12∙20
= £4,700 + £240
= £4,940

Example 4

The first five numbers (terms) in a sequence are 8, 11, 14, 17 and 20.

  1. Write down a formula that allows us find the n-th term of the sequence in terms of the first term without having need to work them out one by one.
  2. Use this formula to find the 23th term of the given sequence.

Solution 4

  1. From the first terms of the sequence it is obvious that they increase by 3, which means the difference between the successive and the previous term is always 3. Therefore, if we denote a whatever term of this sequence by an, then the previous term is the an - 1-th term. Given that the difference is d = 3, we can write
    an - an - 1 = d
    or
    an - an - 1 = 3
    Given that
    a2 = a1 + d
    a3 = a2 + d = (a1 + d) + d = a1 + 2d
    and so on, we obtain for the n-th term in terms of the first one:
    an = a1 + (n - 1)d
    This is the formula for the n-th term of an arithmetic sequence, for which we will discuss more extensively in chapter 12.
    Substituting a1 = 8 and d = 3 yields for this specific case
    an = 8 + 3(n-1)
    = 8 + 3n - 3
    = 3n - 5
  2. Given the above formula, we obtain for the 23th term of this sequence (n = 23)
    an = 3n - 5
    a23 = 3 ∙ 23 - 5
    = 69 - 5
    = 64

From the above example, it is easy to identify the steps one has to follow when dealing with formulas:

Step 1: Identify the variables that are present in the situation under consideration.

Step 2: Assign a different letter to every variable including the one you are going to calculate.

Step 3: Write down the correct formula by including all variables identified in the first step.

Step 4: Substitute the other variables with numbers and calculate the value of the variable you are interested in by using the BEDMAS (PEMDAS) rule.

Example 5

The formula of position x in respect to the time elapsed t for an object moving in a uniformly accelerated motion way is

x(t) = 5 + 3t + 2t2
  1. Calculate the initial position of the object.
  2. Calculate the position of the object after 20 s of motion.

Solution 5

  1. The initial position is obtained for t = 0. Thus,
    x(t) = 5 + 3t + 2t2
    x(0) = 5 + 3 ∙ 0 + 2 ∙ 02
    = 5 + 0 + 0
    = 5
    This means the object was initially 5 m ahead of the origin.
  2. The position after 20 s is calculated by taking t = 20. Thus,
    x(t) = 5 + 3t + 2t2
    x(20) = 5 + 3 ∙ 20 + 2 ∙ 202
    = 5 + 3 ∙ 20 + 2 ∙ 400
    = 5 + 60 + 800
    = 865
    Hence, after 20 s of motion the object will be 865 m ahead of the origin.

Substituting in Formulas derived from Wordy Problems

Wordy problems contain the necessary information to produce one or more formula; you just have to identify the variables participating in the event. This is similar to the mining process used to extract minerals from the Earth core. Then, you can write the equations and apply the same procedure as the one explained earlier.

Example 6

The ratio of Laura's trophies to Anne's trophies is 7:4. The difference between the numbers is 12. What are the numbers?

Solution 6

This is a problem involving ratios, for which we have discussed in chapter 4. We can use a common factor k to reduce the number of variables in the formula. Thus, instead of writing

L:A = 7:4

where L is for Laura and A for Anne, we write instead

L/7 = A/4 = k

so,

L = 7k and A = 4k

In addition, we have

L - A = 12

Therefore, the formula used in this problem is

7k - 4k = 12

It helps calculate the common factor k, i.e.

3k = 12
k = 4

Now, we can find the number of trophies won by each girl. Thus, for Laura, we have

L = 7k = 7 ∙ 4 = 28

and for Anne, we have

A = 4k = 4 ∙ 4 = 16

Let's consider another wordy problem including banking.

Example 7

A customer deposits $15,000 in a bank that applies compound interest rates. If the interest rate is 0.8% and it is compound twice a year, calculate the total amount the customer will have in his savings account after four years.

Solution 7

We have explained the compound interest formula

An (t) = P ∙ (1 + r/n)n ∙ t

in tutorial 5.4, where

An (t) → the amount of deposit after t years;

P → principal (original deposit);

r → interest rate expressed as a decimal (here r = 0.8% = 0.8/100 = 0.008);

n → number of compounds in a year; and

t → time of deposit (in years)

In our example, we have the following values:

P = 15,000
r = 0.008
n = 2
t = 4

In this way, we obtain for the amount the customer will have in his savings account after four years:

An (4) = 15,000 ∙ (1 + 0.008/2)2∙4
= 15,000 ∙ (1 + 0.004)8
= 15,000 ∙ 1.0048
= 15,000 ∙ 1.03245
= 15,486.75

Thus, the customer will have $15,486.75 in his savings account after four years of deposit.

Finding the Inverse Formula

Sometimes, the independent variable in a formula is known and the value of one of dependent variables is required. In this case, we need to find a new formula based on the original one but which expresses the new unknown quantity in terms of the known ones. This is known as the inverse formula. For example, in the formula

y = 3x - 5

we may have the y known and berequired to calculate x. In this case, we isolate x (as unknown) and write the rest of formula in terms of y. Thus, we have

y = 3x - 5
y + 5 = 3x
x = y + 5/3

For example, if y = 7 and we need to find x, we obtain

x = y + 5/3
= 12/3
= 4

Example 8

The number N of dust particles per cubic centimetre present in the environment at an industrial site in terms of the raining time t (in minutes) is given by the formula

N = 80,000/t2

Calculate the raining time necessary to reduce the air pollution to 8 particles per cubic centimetre.

Solution 8

We already know N but we need to calculate the time t. Thus, we isolate t to express it in terms of N. We have

N = 80,000/t2
t2 = 80,000/N
t = √80,000/N

Given that N = 8, we obtain

t = √80,000/8
= √10,000
= 100

Hence, if it rains ceaselessly for 100 minutes in the given site, the air pollution reduces to 8 particles per cubic centimetre.

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