A pencil is of length 13.5cm and a pen is of length 15.5cm. What is the ratio of the length of the pencil to the length of the pen?



What fraction of the length of the pen is the difference between their lengths?

### 1. Ratio of the length of the pencil to the length of the pen:
To find the ratio of the length of the pencil to the length of the pen, we divide the length of the pencil by the length of the pen.

\[
\text{Ratio} = \frac{\text{Length of pencil}}{\text{Length of pen}} = \frac{13.5 \, \text{cm}}{15.5 \, \text{cm}}
\]

Simplifying the fraction:

\[
\frac{13.5}{15.5} = \frac{135}{155}
\]

Now simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 5:

\[
\frac{135}{155} = \frac{135 \div 5}{155 \div 5} = \frac{27}{31}
\]

So, the ratio of the length of the pencil to the length of the pen is:

\[
\boxed{\frac{27}{31}}
\]

### 2. Fraction of the length of the pen that is the difference between their lengths:
First, find the difference between the lengths of the pencil and the pen:

\[
\text{Difference} = 15.5 \, \text{cm} - 13.5 \, \text{cm} = 2 \, \text{cm}
\]

Now, express this difference as a fraction of the length of the pen:

\[
\text{Fraction} = \frac{\text{Difference}}{\text{Length of pen}} = \frac{2}{15.5}
\]

Simplify this fraction:

\[
\frac{2}{15.5} = \frac{2}{15.5} \times \frac{10}{10} = \frac{20}{155}
\]

Now simplify the fraction by dividing both the numerator and denominator by 5:

\[
\frac{20}{155} = \frac{20 \div 5}{155 \div 5} = \frac{4}{31}
\]

So, the fraction of the length of the pen that is the difference between their lengths is:

\[
\boxed{\frac{4}{31}}
\]

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### Related Questions:
1. If the pencil were 2 cm longer, what would the new ratio be?
2. What is the percentage difference in length between the pencil and the pen?
3. What would the difference in lengths be if both the pencil and pen were scaled by a factor of 2?
4. If another pen is 17 cm long, what is the new ratio between the pencil and this new pen?
5. How much longer is the pen compared to the pencil, as a percentage of the pencil's length?

**Tip**: When simplifying fractions, always check for the greatest common divisor to reduce it to its simplest form.

A pencil is of length 13.5cm and a pen is of length 15.5cm. What is the ratio of the length of the pencil to the length of the pen? What fraction of the length of the pen is the difference between their lengths?

Learn how to calculate the ratio of the length of a pencil (13.5 cm) to a pen (15.5 cm) and determine the fraction of the pen's length that represents the difference between their lengths. A step-by-step guide simplifying both ratios and fractions for Grades 6-8 students.

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