Two numbers are less than a third number by 25 % and 30 % , respectively. How much per cent is the second number less than first number ?

Options

A. `6(1)/(3)` %
B. `6(1)/(4)` %
C. `6(2)/(3)` %
D. `6(3)/(4)` %

Answer

To solve the problem step by step, let's define the variables and follow the logic outlined in the video transcript.

### Step 1: Define the Variables
Let:
- \( a \) = First number
- \( b \) = Second number
- \( c \) = Third number

### Step 2: Express the First Number in Terms of the Third Number
According to the problem, the first number \( a \) is 25% less than the third number \( c \). We can express this mathematically as:
\[ a = c - 0.25c = 0.75c \]
This means:
\[ a = \frac{3}{4}c \]

### Step 3: Express the Second Number in Terms of the Third Number
Similarly, the second number \( b \) is 30% less than the third number \( c \). This can be expressed as:
\[ b = c - 0.30c = 0.70c \]
Thus:
\[ b = \frac{7}{10}c \]

### Step 4: Find the Relationship Between the First and Second Numbers
Now we need to find how much percent the second number \( b \) is less than the first number \( a \). We can express this as:
\[ \text{Percentage decrease} = \frac{a - b}{a} \times 100 \]

### Step 5: Substitute the Values of \( a \) and \( b \)
Substituting the expressions for \( a \) and \( b \):
\[ \text{Percentage decrease} = \frac{0.75c - 0.70c}{0.75c} \times 100 \]

### Step 6: Simplify the Expression
Now, simplify the numerator:
\[ 0.75c - 0.70c = 0.05c \]
So, we have:
\[ \text{Percentage decrease} = \frac{0.05c}{0.75c} \times 100 \]
The \( c \) cancels out:
\[ \text{Percentage decrease} = \frac{0.05}{0.75} \times 100 \]

### Step 7: Calculate the Final Value
Now, calculate \( \frac{0.05}{0.75} \):
\[ \frac{0.05}{0.75} = \frac{5}{75} = \frac{1}{15} \]
Thus:
\[ \text{Percentage decrease} = \frac{1}{15} \times 100 = \frac{100}{15} = 6.67\% \]

### Step 8: Final Result
The second number is approximately \( 6.67\% \) less than the first number.

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