Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Siyavula textbooks: Grade 10 Maths [CAPS] » Mathematical models

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • FETMaths display tagshide tags

    This module and collection are included inLens: Siyavula: Mathematics (Gr. 10-12)
    By: Siyavula

    Module Review Status: In Review
    Collection Review Status: In Review

    Click the "FETMaths" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Equations and inequalities: Mathematical models

Introduction

Tom and Jane are friends. Tom picked up Jane's Physics test paper, but will not tell Jane what her marks are. He knows that Jane hates maths so he decided to tease her. Tom says: 'I have 2 marks more than you do and the sum of both our marks is equal to 14. How much did we get?'

Let's help Jane find out what her marks are. We have two unknowns, Tom's mark (which we shall call tt) and Jane's mark (which we shall call jj). Tom has 2 more marks than Jane. Therefore,

t = j + 2 t = j + 2
(1)

Also, both marks add up to 14. Therefore,

t + j = 14 t + j = 14
(2)

The two equations make up a set of linear (because the highest power is one) simultaneous equations, which we know how to solve! Substitute for tt in the second equation to get:

t + j = 14 j + 2 + j = 14 2 j + 2 = 14 2 ( j + 1 ) = 14 j + 1 = 7 j = 7 - 1 = 6 t + j = 14 j + 2 + j = 14 2 j + 2 = 14 2 ( j + 1 ) = 14 j + 1 = 7 j = 7 - 1 = 6
(3)

Then,

t = j + 2 = 6 + 2 = 8 t = j + 2 = 6 + 2 = 8
(4)

So, we see that Tom scored 8 on his test and Jane scored 6.

This problem is an example of a simple mathematical model. We took a problem and we were able to write a set of equations that represented the problem mathematically. The solution of the equations then gave the solution to the problem.

Problem Solving Strategy

The purpose of this section is to teach you the skills that you need to be able to take a problem and formulate it mathematically in order to solve it. The general steps to follow are:

  1. Read ALL of the question !
  2. Find out what is requested.
  3. Use a variable(s) to denote the unknown quantity/quantities that has/have been requested e.g., xx.
  4. Rewrite the information given in terms of the variable(s). That is, translate the words into algebraic expressions.
  5. Set up an equation or set of equations (i.e. a mathematical sentence or model) to solve the required variable.
  6. Solve the equation algebraically to find the result.

Application of Mathematical Modelling

Exercise 1: Mathematical Modelling: Two variables

Three rulers and two pens have a total cost of R 21,00. One ruler and one pen have a total cost of R 8,00. How much does a ruler costs on its own and how much does a pen cost on its own?

Solution
  1. Step 1. Translate the problem using variables :

    Let the cost of one ruler be xx rand and the cost of one pen be yy rand.

  2. Step 2. Rewrite the information in terms of the variables :
    3 x + 2 y = 21 x + y = 8 3 x + 2 y = 21 x + y = 8
    (5)
  3. Step 3. Solve the equations simultaneously :

    First solve the second equation for yy:

    y = 8 - x y = 8 - x
    (6)

    and substitute the result into the first equation:

    3 x + 2 ( 8 - x ) = 21 3 x + 16 - 2 x = 21 x = 5 3 x + 2 ( 8 - x ) = 21 3 x + 16 - 2 x = 21 x = 5
    (7)

    therefore

    y = 8 - 5 y = 3 y = 8 - 5 y = 3
    (8)
  4. Step 4. Present the final answers :

                One ruler costs R 5,00 and one pen costs R 3,00.

Exercise 2: Mathematical Modelling: One variable

A fruit shake costs R2,00 more than a chocolate milkshake. If three fruit shakes and 5 chocolate milkshakes cost R78,00, determine the individual prices.

Solution
  1. Step 1. Summarise the information in a table :

    Let the price of a chocolate milkshake be xx and the price of a fruitshake be yy.

    Table 1
      Price number Total
    Fruit y y 3 3 y 3 y
    Chocolate x x 5 5 x 5 x
  2. Step 2. Set up a pair of algebraic equations :
    3 y + 5 x = 78 3 y + 5 x = 78
    (9)

    y=x+2y=x+2

  3. Step 3. Solve the equations :
    3 ( x + 2 ) + 5 x = 78 3 x + 6 + 5 x = 78 8 x = 72 x = 9 y = x+2 = 9 + 2 = 11 3 ( x + 2 ) + 5 x = 78 3 x + 6 + 5 x = 78 8 x = 72 x = 9 y = x+2 = 9 + 2 = 11
    (10)
  4. Step 4. Present the final answer :

    One chocolate milkshake costs R 9,00 and one Fruitshake costs R 11,00

Mathematical Models

  1. Stephen has 1 l of a mixture containing 69% of salt. How much water must Stephen add to make the mixture 50% salt? Write your answer as a fraction of a litre.
     
    Click here for the solution
  2. The diagonal of a rectangle is 25 cm more than its width. The length of the rectangle is 17 cm more than its width. What are the dimensions of the rectangle?
     
    Click here for the solution
  3. The sum of 27 and 12 is 73 more than an unknown number. Find the unknown number.
     
    Click here for the solution
  4. The two smaller angles in a right-angled triangle are in the ratio of 1:2. What are the sizes of the two angles?
     
    Click here for the solution
  5. George owns a bakery that specialises in wedding cakes. For each wedding cake, it costs George R150 for ingredients, R50 for overhead, and R5 for advertising. George's wedding cakes cost R400 each. As a percentage of George's costs, how much profit does he make for each cake sold?
     
    Click here for the solution
  6. If 4 times a number is increased by 7, the result is 15 less than the square of the number. Find the numbers that satisfy this statement, by formulating an equation and then solving it.
     
    Click here for the solution
  7. The length of a rectangle is 2 cm more than the width of the rectangle. The perimeter of the rectangle is 20 cm. Find the length and the width of the rectangle.
     
    Click here for the solution

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks