Question

1. andre took 6 kicks and made 4 goals: \\(\frac{4}{6}\\). tom wants to take more than 20 kicks and make an equivalent fraction to andre. what are two possible fractions of goals tom can make?

Explanation:

Step1: Simplify Andre's fraction

First, simplify the fraction \(\frac{4}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
\(\frac{4\div2}{6\div2}=\frac{2}{3}\)

Step2: Find equivalent fractions

To find equivalent fractions, we can multiply the numerator and the denominator of \(\frac{2}{3}\) by the same non - zero integer.

• For multiplying by 2:

\(\frac{2\times2}{3\times2}=\frac{4}{6}\) (this is the original fraction, but we can also get new ones)

• For multiplying by 3:

\(\frac{2\times3}{3\times3}=\frac{6}{9}\)

• For multiplying by 4:

\(\frac{2\times4}{3\times4}=\frac{8}{12}\)
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If Tom takes more than 20 kicks, let's assume the number of kicks \(n>20\). Let the number of goals be \(g\). We know that \(\frac{g}{n}=\frac{2}{3}\), so \(g = \frac{2}{3}n\). Since \(n\) must be a multiple of 3 (because \(g\) must be an integer), let's take \(n = 21\) (the smallest multiple of 3 greater than 20). Then \(g=\frac{2}{3}\times21 = 14\), and the fraction is \(\frac{14}{21}\). Another one: if \(n = 24\), then \(g=\frac{2}{3}\times24 = 16\), and the fraction is \(\frac{16}{24}\)

Answer:

Two possible fractions are \(\frac{6}{9}\) and \(\frac{8}{12}\) (or \(\frac{14}{21}\), \(\frac{16}{24}\) etc. when considering more than 20 kicks).