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Mathematics LibreTexts

11.2: Basic Terminology

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Definitions:

Consecutive integers (specific numbers): 7, 8, 9.

Consecutive integers (general numbers in a box): x, x+1, x+2.

Consecutive even integers (specific numbers): 8, 10, 12.

Consecutive even integers (general numbers in a box): x, x+2, x+4.

Consecutive odd integers (specific numbers): 7, 9, 11

Consecutive integers (general numbers in a box): x, x+2, x+4.

Examples

Some of the problems here are simple. The solution can be worked out fast by quick reasoning. The benefit of these problems is not to find the solution by reasoning, but by learning algebraic steps applicable to more challenging situations.

The general method is not to read a problem first and understand it. Many of my colleagues will disagree with me. I propose creating a preamble in which you write a symbol or set of symbols for each element in the problem. (Refer to exercise 10.2.9).

Look at the end of the problem where you usually find what to solve for. Let x be the number we are looking for. Write this as the first step in the preamble (list of symbols in non-trivial exercises whose solution we seek.)

Next start reading the problem from the beginning and develop the preamble step by step. Write the symbol(s) for each step on a new line.

Finish the preamble by converting all the steps in the problem into symbols. Now look at your preamble and gain an overview of the your problem.

It should be easier to obtain an equation using the symbols from the preamble. Then solve the equation by the method introduced this far for linear equations. (A linear equation has a variable to the first degree (exponent).)

Example 11.2.1

The sum of three consecutive positive integers is 705. Find the integers.

Solution:

Preamble:

Let x be the smallest of the three consecutive integers (for example x=20).
Then x+1 is the middle number 20+1=21.
And x+2 is the largest number 20+2=22.
The sum of three integers is x+(x+1)+(x+2)=3x+3.

Equation:

Sum of integers=sum of integers3x+3=7053x+33=70533x=7023x3=7023x=234

Smallest integer: x=234
Middle integer: x+1=234+1=235
largest integer: x+2=234+2=236

Preamble:

Let x be the middle of the three consecutive integers (for example x=20).
[5pt] Then x1 is the smallest number 201=19.
And x+1 is the largest number 20+1=21.
The sum of three integers is (x1)+(x+1)+(x)+(x+1)=3x.
Equation: Sum of integers=sum of integers3x=7053x=7053x3=7053x=235

Smallest integer: x1=234
Middle integer: x=235
largest integer: x+1=234+2=236

Preamble:

Let x be the largest of the three consecutive integers (for example x=20).
[5pt] Then x1 is the middle number 20+1=19.
And x2 is the smallest number 20+2=18.
The sum of three integers is (x2)+(x1)+x=3x3.
Equation: Sum of integers=sum of integers3x3=7053x+33=705+33x=7083x3=7083x=236

Smallest integer: x=234
Middle integer: x+1=234+1=235
largest integer: x+2=234+2=236

Example 11.2.3

Find two integers whose sum is 82.

11 more than three times the smaller number is the same as 18 less than twice the larger number.
Find the numbers.

Solution:

Preamble:

Let x be the smaller number (for example x=20).
Then the larger number is 82x 8220.
11 more than 3× the smaller number: 3x+11 3(20)+11.
18 less than 2×
the larger number: 2(82x)18=1642x18=1462x 2(8220)18).
Equation:

\boldsymbol{\begin{array}{rcl lll} 11\hbox{ more than three times the smaller number}&=&\hbox{\)18\(less than twice the larger number}\\[6pt] 3x+11&=&146-2x\\[6pt] 3x+2x+11&=&146-2x+2x\\[6pt] 5x+11&=&146\\[6pt] 5x+11-11&=&146-11\\[6pt] 5x&=&135\\[10pt] \displaystyle \frac{5}{5}x&=&\displaystyle \frac{135}{5}\\[5pt] x&=&27 \end{array}}

The smaller number is x=27.
The larger number is 8227=55.

Preamble:

Let x be the larger number (for example x=20).
Then the smaller number is 82x 8220.
[5pt] 11 more than 3× 3(82x)+11 3(8220)+11.
smaller number: =246+113x .
=2573x .
18 less than 2× the larger number: 2(x)18 2(20)18).

Equation: 11morethan3×smaller=18lessthantwicelarger2573x=2x182573x+3x=2x+3x18257=5x18257+18=5x18+18275=5xx=2755x=55 The smaller number is 8255=27.
The larger number is 55.

Example 11.2.3

Find two consecutive odd integers such that 60 less than three times the larger number equals 73 more than twice the smaller number.

Solution

Preamble:

Let x be the smaller odd number (for example x=21).
Then larger number is x+2 21+2.
[5pt] 60 less than 3×3(x+2)60 3(21+2)60.
the larger number: =3x+660 .
=3x54 .
73 more than 2× smaller: 2x+73 2(21)+73.
Equation: 60 less than 3 times larger=text73morethan2×smallernumber3x54=2x+733x54+54=2x+73+543x=2x+1273x2x=2x2x+127x=127 The smaller number is x=127.
The larger number is 127+2=129.

Preamble:

Let x be the larger odd number (for example x=21).
Then the smaller number is x2 212.
[5pt] 60 less than 3× the larger number: 3x60 3(21)60 .
73 more than 2× smaller: 2(x2)+73 2(212)+73.
=2x4+73 .
=2x+69 .
Equation: 60 less than 3 \times larger=73 more than 2\times smaller number3x60=2x+693x60+60=2x+69+603x=2x+1293x2x=2x2x+129x=129 The smaller number is x=1292=127.
The larger number is 129.

Exercises 11

  1. The sum of three consecutive positive integers is 1,128. Find the integers.
  2. Find two consecutive even integers such that 150 less than three times the smaller number equals 148 more than twice the larger number.
  3. Find two integers whose sum is 425.
    10 more than six times the smaller number is the same twice the larger number.
    Find the numbers.
  1. The sum of three consecutive positive integers is 1,128. Find the integers.

    Solution:

    Preamble:

    Let x be the smallest of the three consecutive integers (for example x=20).
    [5pt] Then x+1 is the middle number 20+1=21.
    And x+2 is the largest number 20+2=22.
    The sum of three integers is x+(x+1)+(x+2)=3x+3.
    Equation: Sum of integers=sum of integers3x+3=1,1283x+33=1,12833x=1,1253x3=1,1253x=375

    Smallest integer: x=375
    Middle integer: x+1=375+1=376
    largest integer: x+2=375+2=377

  2. Find two consecutive even integers such that 150 less than three times the smaller number equals 148 more than twice the larger number.

    Solution:

    Preamble:

    Let x be the smaller number (for example x=21).
    Then the larger even number is x+2 21+2.
    150 less than 3× the smaller number: 3x150 3(21)150.
    148 more than 2×
    the larger number: 2(x+2)+148=2x+4+148=2x+152 2(21+2)+148).

    Equation:
    \boldsymbol{\begin{array}{rcl lll} 150\hbox{ less than\)3\(smaller number}&=&\hbox{\)148\(more than\)2\(larger number}\\[6pt] 3x-150&=&2x+152\\[6pt] 3x-2x-150&=&2x-2x+152\\[6pt] x-150&=&152\\[6pt] x-150+150&=&152+150\\[6pt] x&=&302\\[10pt] \end{array}}

    The smaller number is x=302.
    The larger number is 302+2=304.

  3. Find two integers whose sum is 425.
    10 more than six times the smaller number is the same as twice the larger number.
    Find the numbers.

    Solution:

    Preamble:

    Let x be the smaller number (for example x=20).
    Then the larger number is 425x 42520.
    [5pt] 10 more than 6× smaller number: 6x+10 6(20)+10.
    10 more than 6× smaller: 6x+10 6(20)+10.

    Equation: 10 more than 6 \times smaller=73 more than 2\times larger number6x+10=8502x6x+2x+10=8502x+2x8x+10=8508x+1010=850108x=84088x=8408x=105 The smaller number is x=105.
    The larger number is 425105=320.


11.2: Basic Terminology is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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