**Solving Word Problems with Three Variables :**

In this section will learn, how to solve word problems with three variables.

**Example 1 :**

Vani, her father and her grand father have an average age of 53. One-half of her grand father’s age plus one-third of her father’s age plus one fourth of Vani’s age is 65. Four years ago if Vani’s grandfather was four times as old as Vani then how old are they all now ?

**Solution :**

Let x, y and z be the ages of Vani, her father, her grand father respectively.

**Given :** Average age of them is 53.

Then, we have

(x + y + z) / 3 = 53

Multiply each side by 3.

x + y + z = 159 -----(1)

**Given :** One-half of her grand father’s age plus one-third of her father’s age plus one fourth of Vani’s age is 65.

Then, we have

(1/2)z + (1/3)y + (1/4)x = 65

(x/4) + (y/3) + (z/2) = 65

L.C.M of (2, 3, 2) is 12. So, multiply each side by 12.

3x + 4y + 6z = 65(12)

3x + 4y + 6z = 780 -----(2)

**Given :** Four years ago, Vani’s grandfather was four times as old as Vani.

Then, we have

z - 4 = 4(x - 4)

z - 4 = 4x - 16

4x - z = 16 - 4

4x - z = 12 -----(3)

In order to eliminate y, subtract (2) from 4 times of (1).

4(1) - (2) :

- x + 2z = - 144 -----(4)

Add 2 times of (3) and (4).

2(3) + (4) :

7x = 168

Divide each side by 7.

x = 24

Substitute 24 for x in (3).

(3)-----> 4(24) - z = 12

96 - z = 12

Subtract 96 from each side.

- z = - 84

Multiply each side by (-1).

z = 84

Substitute 24 for x and 84 for y in (1).

(1)-----> x + y + z = 159

24 + y + 84 = 159

y + 108 = 159

Subtract 108 from each side.

y = 51

The values of x, y and z are

x = 24

y = 51

z = 84

So, Vani is 24 years old, his father is 51 years old and and his grandfather is 84 years old.

**Example 2 :**

The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five times the former number. If the hundreds digit plus twice the tens digit is equal to the units digit, then find the original three digit number ?

**Solution :**

Let x, y and z be the digits in hundreds place, tens place and one place in the three digit number.

Then, the required three digit number is

xyz

**Given :** The sum of the digits 11.

Then, we have

x + y + z = 11 -----(1)

**Given :** If the digits are reversed, the new number is 46 more than five times the former number.

Then, we have

zyx = 5(xyz) + 46

100z + 10y + 1x = 5(100x + 10y + z) + 46

Simplify.

100z + 10y + 1x = 500x + 50y + 5z + 46

-499x - 40y + 95z = 46 -----(2)

**Given :** If the hundreds digit plus twice the tens digit is equal to the units digit.

Then, we have

x + 2y = z

x + 2y - z = 0 -----(3)

In order to eliminate y in (1) and (2), subtract (2) from 40 times (1).

40(1) - (2) :

- 459x + 135z = 486 -----(4)

In order to eliminate y in (1) and (3), subtract (3) from 2 times (1).

2(1) - (3) :

x + 3z = 22 -----(5)

In order to eliminate z in (4) and (5), subtract 45 times of (5) from (4).

(4) - 45(5) :

- 504x = - 504

Divide each side by -504.

x = 1

Substitute 1 for x in (5).

(5)-----> 1 + 3z = 22

Subtract 1 from each side.

3z = 21

Divide each side by 3.

z = 7

Substitute 1 for x and 7 for z in (1).

(1)-----> 1 + y + 7 = 11

y + 8 = 11

Subtract 8 from each side.

y = 3

The values of x, y and z are

x = 1

y = 3

z = 7

xyz = 137

So, the required three digit number is 137.

**Example 3 :**

There are 12 pieces of five, ten and twenty dollar currencies whose total value is $105. When first 2 sorts are interchanged in their numbers its value will be increased by $20. Find the number of currencies in each sort.

**Solution :**

Let x, y and z be the number of pieces of five, ten, and twenty rupee currencies.

x + y + z = 12 -----(1)

**Given : **The total value of the currencies is $108.

Then, we have

5x + 10y + 20z = 105

Divide each side by 5.

x + 2y + 4z = 21 -----(2)

**Given : **When first 2 sorts are interchanged in their numbers, its value will be increased by $20.

Then, we have

10x + 5y + 20z = 105 + 20

Simplify.

10x + 5y + 20z = 125

Divide each side by 5.

2x + y + 4z = 25 -----(3)

In order to eliminate z in (1) and (2), subtract (2) from 4 times of (1).

4(1) - (2) :

3x + 2y = 27 -----(4)

In order to eliminate z in (2) and (3), subtract (3) from (2).

(2) - (3) :

- x + y = - 4 -----(5)

In order to eliminate x in (4) and (5), add (4) and 3 times of (5).

(4) + 3(5) :

5y = 15

Divide each side by 5.

y = 3

Substitute 3 for y in (4).

(4)-----> 3x + 2(3) = 27

3x + 6 = 27

Subtract 6 from each side.

3x = 21

Divide each side by 3.

x = 7

Substitute 7 for x and 3 for y in (1).

(1)-----> 7 + 3 + z = 12

10 + z = 12

Subtract 10 from each side.

z = 2

The values of x, y and z are

x = 7

y = 3

z = 2

So, the number of 5 rupee note is 7, number of ten rupee note is 3 and number of twenty rupee note is 2.

After having gone through the stuff given above, we hope that the students would have understood, how to solve word problems involving linear equations in three variables.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**