Maharashtra Board Class 5 Maths Part One Chapter 5 Fractions PDF Download

Fractions

Equivalent Fractions

If one bhakari is divided equally between two people, each one will get half a bhakari. The fraction half is written as \(\frac{1}{2}\). Here 1 is the numerator and 2 is the denominator.

One bhakari was divided into four equal parts. Two of the parts were given away. This is shown as \(\frac{2}{4}\). Here, 2 is the numerator and 4, the denominator. This, too, means that half a bhakari was given.

Six equal parts were made of one melon. They were shared equally by two people. It means that the part that each one got was \(\frac{3}{6}\). Each one actually got half the melon. Thus, \(\frac{3}{6}\) also shows the fraction 'one half'.

In the three examples above, the fraction 'half' has been shown by \(\frac{1}{2}\), \(\frac{2}{4}\), \(\frac{3}{6}\) respectively. It means that the value of all three fractions is the same. This is written as \(\frac{1}{2} = \frac{2}{4} = \frac{3}{6}\).

Such fractions of equal value are called equivalent fractions.

Look at the coloured parts of the two equal circles shown alongside. One circle is divided into 3 equal parts and two of them are coloured. That is, the coloured part is \(\frac{2}{3}\) of the circle.

The other circle of the same size is divided into six equal parts and 4 of them are coloured. That is, \(\frac{4}{6}\) of the whole circle is coloured. However, we see that the coloured parts of the two circles are equal. Therefore, \(\frac{2}{3} = \frac{4}{6}\).

Thus, \(\frac{2}{3}\) and \(\frac{4}{6}\) are equivalent fractions.

Teacher's Note

When you buy half a kg of sugar and your friend buys 500 grams, you both get the same amount. These are equivalent fractions in real life.

Exam Trick

Remember: If you multiply or divide both numerator and denominator by the same number, the fraction stays the same. This is the golden rule for equivalent fractions.

Points to Remember

Equivalent fractions have the same value but different numerators and denominators.
You can make equivalent fractions by multiplying the numerator and denominator by the same number.
You can also divide the numerator and denominator by the same number to get equivalent fractions.
\(\frac{1}{2} = \frac{2}{4} = \frac{3}{6}\) are all equivalent fractions.
Always look for a common factor to find equivalent fractions.

Obtaining Equivalent Fractions

Two of the 5 equal parts in the figure are coloured. The coloured part is \(\frac{2}{5}\) of the whole figure.

When two lines are drawn across the same figure, it gets divided into 15 equal parts. So, now, the fraction that shows the coloured part is \(\frac{6}{15}\).

However, the coloured part has not changed. Therefore, we see that \(\frac{2}{5} = \frac{6}{15}\).

Teacher: Do you see any special connection between the numerators and denominators of the fractions \(\frac{2}{5}\) and \(\frac{6}{15}\)?

Sonu: Three times 2 is 6 and three times 5 is fifteen.

Teacher: We have also seen that \(\frac{1}{2} = \frac{2}{4}\), \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{2}{3} = \frac{4}{6}\). In two equivalent fractions, the numerator of one fraction is as many times the numerator of the other as the denominator of one is of the denominator of the other.

When the numerator and denominator of a fraction are multiplied by the same non-zero number, we get a fraction that is equivalent to the given fraction.

Nandu: Can I get an equivalent fraction by dividing the numerator and denominator by the same number?

Teacher: Of course! If the numerator and denominator have a common divisor, then the fraction obtained on actually dividing them by that divisor is equivalent to the given fraction. The numerator and denominator of the fraction \(\frac{6}{15}\) can be divided by 3. On doing this division, we get the fraction \(\frac{2}{5}\).

It means that \(\frac{6}{15} = \frac{2}{5}\).

If the numerator and denominator have a common divisor then the fraction we get on dividing them by that divisor is equivalent to the given fraction.

Teacher: Divide the numerator and denominator of \(\frac{6}{12}\) by the same number to find an equivalent fraction.

Nandu: 6 and 12 can also be divided by 6. Will that do?

Teacher: Sure. \(\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}\).

Remember that the fractions we get by dividing \(\frac{6}{12}\) by 2 or 3 or 6 are all equivalent to \(\frac{6}{12}\). That is \(\frac{6}{12} = \frac{3}{6} = \frac{2}{4} = \frac{1}{2}\).

Sonu obtained this fraction: \(\frac{6}{12} = \frac{6 \div 2}{12 \div 2} = \frac{3}{6}\)

Minu obtained this fraction: \(\frac{6}{12} = \frac{6 \div 3}{12 \div 3} = \frac{2}{4}\)

Example (1) Find a fraction with denominator 30 which is equivalent to \(\frac{5}{6}\).

\(\frac{5}{6} = \frac{\square}{30}\). We must find the right number for the box.

Here, 5 times the denominator 6 is 30. What is five times the numerator 5?

\(\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}\). Hence, the fraction \(\frac{25}{30}\) with denominator 30 is equivalent to \(\frac{5}{6}\).

Teacher's Note

When you divide a pizza, \(\frac{2}{4}\) and \(\frac{1}{2}\) are the same amount. Both give you one whole half of the pizza.

Exam Trick

To find an equivalent fraction, look at what you multiply the denominator by. Then multiply the numerator by the same number. For example, if 6 becomes 30, you multiplied by 5. So multiply 5 by 5 also.

Points to Remember

Multiply both top and bottom by the same number to get equivalent fractions.
Divide both top and bottom by the same number to get equivalent fractions.
The coloured part does not change when you find equivalent fractions.
Always multiply or divide both numbers, not just one.
Check your answer: the new fraction should show the same amount as the old one.

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