**Equations with rational numbers :**

In this section, we are going to see, how to solve an equation with variables on both sides that involves rationals numbers (fractions and decimals).

**Example 1 :**

Solve : 7n/10 + 3/2 = 3n/5 + 2

**Solution :**

**Step 1 : **

Find the least common multiple of the denominators (10, 2 and 5).

LCM = 5 x 2 x 1 x 1 x 1

LCM = 10

**Step 2 : **

Multiply both sides of the equation by 10.

10(7n/10 + 3/2) = 10(3n/5 + 2)

Use distributive property.

10(7n/10) + 10(3/2) = 10(3n/5) + 10(2)

Simplify.

7n + 5(3) = 2(3n) + 20

7n + 15 = 6n + 20

**Step 3 :**

Use inverse operations to solve for "n".

Subtract 15 from both sides.

aaaaaaaaaaaaaaaa 7n + 15 = 6n + 20 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa - 15 - 15 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa---------------------- aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa 7n = 6n + 5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa---------------------- aaaaaaaaaaaaaaaa

Subtract 6n from both sides.

aaaaaaaaaaaaaaaa 7n = 6n + 5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa -6n -6n aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa -------------------- aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa n = 5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa -------------------- aaaaaaaaaaaaaaaa

**Example 2 :**

Solve : k/7 - 6 = 3k/7 + 4

**Solution :**

**Step 1 : **

Find the least common multiple of the denominators.

Here, we have the same denominator on both sides. That is 7.

So, LCM = 7.

**Step 2 : **

Multiply both sides of the equation by 7.

7(k/7 - 6) = 7(3k/7 + 4)

Use distributive property.

7(k/7) - 7(6) = 7(3k/7) + 7(4)

Simplify.

k - 42 = 3k + 28

**Step 3 :**

Use inverse operations to solve for "k".

Subtract 28 from both sides.

aaaaaaaaaaaaaaaa k - 42 = 3k + 28 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa - 28 - 28 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa---------------------- aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa k - 70 = 3k aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa---------------------- aaaaaaaaaaaaaaaaaa

Subtract k from both sides.

aaaaaaaaaaaaaaaaaaaa k - 70 = 3k aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa - k - k aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa---------------- aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa -70 = 2k aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa---------------- aaaaaaaaaaaaaaaaaa

Divide both sides by 2.

-70 / 2 = 2k / 2

-35 = k

**Example 3 :**

Solve : 0.7n + 0.33 = 0.3n + 0.5

**Solution :**

**Step 1 :**

In the second term 0.33 on the left side, we have two digits (more number of digits) after the decimal.

So, multiply both sides of the equation by 10² ( = 100).

100(0.7n + 0.33) = 100(0.3n + 0.5)

100(0.7n) + 100(0.33) = 100(0.3n) + 100(0.5)

70n + 33 = 30n + 50

**Step 2 :**

Subtract 33 from both sides.

aaaaaaaaaaaaaaaa 70n + 33 = 30n + 50 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa - 33 - 33 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa ------------------------ aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa 70n = 30n + 17 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa ------------------------ aaaaaaaaaaaaaaaa

Subtract 30n from both sides.

aaaaaaaaaaaaaaaa 70n = 30n + 17 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaa - 30n = -30n aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa------------------------- aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa 40n = 17 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa------------------------- aaaaaaaaaaaaaaaa

Divide both sides by 40.

40n/40 = 17/40

n = 0.425

**Example 4 :**

David walks from his house to the zoo at a constant rate. After walking 0.75 mile, he meets his brother Daniel, and they continue walking at the same constant rate. When they arrive at the zoo, David has walked for 0.5 hour and Daniel has walked for 0.2 hour. What is the rate in miles per hour at which the brothers walked to the zoo ?

**Solution :**

**Step 1 : **

Write an expression for the distance from the brothers’ house to the zoo, using the fact that distance equals rate times time.

Let r be the walking rate of both David and his brother Daniel.

Distance from the brothers’ house to the zoo

= 0.2r

**Step 2 : **

Write an expression for the distance from the David's house to the zoo, using the distance from his brother's house to the zoo.

Distance from Davids’ house to the zoo

= 0.75 + 0.2r -------(1)

**Step 3 :**

Write an expression for the distance from the David's house to the zoo, using David's total walking time 0.5 hour.

Distance from Davids’ house to the zoo

= 0.5r -------(1)

**Step 4 : **

Both (1) and (2) represent the distance from David's house to the zoo.

So, we have

(1) = (2)

0.75 + 0.2r = 0.5r

**Step 5 :**

In the first term 0.75 on the left side, we have two digits (more number of digits) after the decimal.

So, multiply both sides of the equation by 10² ( = 100).

100(0.75 + 0.2r) = 100(0.5r)

100(0.75) + 100(0.2r) = 50r

75 + 20r = 50r

**Step 6 :**

Subtract 20r from both sides.

aaaaaaaaaaaaaaaaaaa 75 + 20r = 50r aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa - 20r -20r aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa ------------------ aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa 75 = 30r aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa ------------------ aaaaaaaaaaaaaaaaa

Divide both sides by 30.

75 / 30 = 30r / 30

2.5 = r

Hence, the brothers’ constant rate of speed was 2.5 miles per hour.

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