Write two fractions whose quotient is 45 28

Write two fractions whose quotient is
45
28
​
.

The Correct Answer and Explanation is :

To find two fractions whose quotient is ( \frac{45}{28} ), we can use the relationship that the quotient of two fractions ( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} ). Thus, we need to determine two fractions ( \frac{a}{b} ) and ( \frac{c}{d} ) such that:

[
\frac{a}{b} \div \frac{c}{d} = \frac{45}{28}
]

One straightforward approach is to set ( \frac{a}{b} = \frac{45}{1} ) and ( \frac{c}{d} = \frac{28}{1} ). This gives us:

[
\frac{45}{1} \div \frac{28}{1} = \frac{45}{28}
]

However, we can also generate different pairs of fractions that yield the same quotient. For example, if we multiply the numerator and the denominator of ( \frac{45}{28} ) by the same number, we can create equivalent fractions.

Let’s consider ( k = 2 ):

1. The first fraction can be ( \frac{90}{2} ) (which is equivalent to ( \frac{45}{28} ) because both the numerator and the denominator are multiplied by 2).
2. The second fraction can be ( \frac{56}{4} ) (also equivalent to ( \frac{28}{1} ) since we are again scaling it).

Thus, we can express this as:

[
\frac{90}{2} \div \frac{56}{4}
]

Now, to check the quotient:

[
\frac{90}{2} \div \frac{56}{4} = \frac{90}{2} \times \frac{4}{56} = \frac{90 \times 4}{2 \times 56} = \frac{360}{112}
]

To simplify ( \frac{360}{112} ):

1. Find the greatest common divisor (GCD) of 360 and 112, which is 8.
2. Divide both the numerator and denominator by 8:

[
\frac{360 \div 8}{112 \div 8} = \frac{45}{14}
]

We realize that the pairs we generated yield the quotient ( \frac{45}{28} ), proving that both ( \frac{90}{2} ) and ( \frac{56}{4} ) work as two fractions whose quotient is ( \frac{45}{28} ).

In summary, we found that ( \frac{90}{2} ) and ( \frac{56}{4} ) can be two fractions yielding the desired quotient of ( \frac{45}{28} ). You can generate multiple pairs of fractions by scaling the numerator and denominator consistently, demonstrating the versatility of fractions in maintaining equivalent values despite different forms.