Question

fractions

1. which of the following inequalities is true for the fractions \\(\frac{2}{5}\\), \\(\frac{5}{13}\\), and \\(\frac{3}{7}\\)?

a \\(\frac{5}{13} < \frac{2}{5} < \frac{3}{7}\\)
b \\(\frac{5}{13} < \frac{3}{7} < \frac{2}{5}\\)
c \\(\frac{2}{5} < \frac{3}{7} < \frac{5}{13}\\)
d \\(\frac{2}{5} < \frac{5}{13} < \frac{3}{7}\\)
how do you convert a fraction to a decimal?
a multiply the denominator by the numerator.
b divide the numerator by the denominator.
c divide the denominator by the numerator.
d nicely ask a friend to do it for you.

Explanation:

First Question (How to convert a fraction to a decimal?)

To convert a fraction to a decimal, we use the operation of division. A fraction \(\frac{a}{b}\) (where \(a\) is the numerator and \(b\) is the denominator, \(b
eq0\)) is converted to a decimal by dividing the numerator by the denominator, i.e., \(a\div b\). Let's analyze each option:

• Option A: Multiplying the denominator by the numerator is not the correct operation for converting a fraction to a decimal. For example, for \(\frac{1}{2}\), multiplying denominator (2) by numerator (1) gives 2, which is not the decimal equivalent (0.5).
• Option B: Dividing the numerator by the denominator is the correct method. For \(\frac{1}{2}\), \(1\div2 = 0.5\), which is the decimal form.
• Option C: Dividing the denominator by the numerator would give the reciprocal's decimal. For \(\frac{1}{2}\), \(2\div1=2\), which is not the decimal for \(\frac{1}{2}\).
• Option D: Asking a friend is not a mathematical method for conversion.

To determine the correct inequality, we convert the fractions \(\frac{2}{5}\), \(\frac{5}{13}\), and \(\frac{3}{7}\) to decimals. \(\frac{5}{13}\approx0.3846\), \(\frac{2}{5} = 0.4\), and \(\frac{3}{7}\approx0.4286\). Comparing these decimals, we see that \(0.3846<0.4<0.4286\), which corresponds to the inequality \(\frac{5}{13}<\frac{2}{5}<\frac{3}{7}\) (Option A). Other options are evaluated as follows:

• Option B: \(\frac{6}{13}\) is not one of the given fractions, so this option is incorrect as it introduces an extra fraction.
• Option C: The order \(0.4<0.4286<0.4615\) (for \(\frac{2}{5}<\frac{3}{7}<\frac{6}{13}\)) is incorrect as \(\frac{6}{13}\) is not a given fraction.
• Option D: The order \(\frac{2}{5}<\frac{5}{13}<\frac{3}{7}\) is incorrect because \(\frac{5}{13}\approx0.3846<\frac{2}{5} = 0.4\), not the other way around.

Answer:

B. Divide the numerator by the denominator.

Second Question (Fraction Inequalities)

To solve this, we can convert the fractions to decimals or find a common denominator to compare them. Let's take each option:

Option A: \(\frac{5}{13}<\frac{2}{5}<\frac{3}{7}\)

• Convert to decimals:
• \(\frac{5}{13}\approx0.385\) (since \(5\div13\approx0.385\))
• \(\frac{2}{5} = 0.4\)
• \(\frac{3}{7}\approx0.429\) (since \(3\div7\approx0.429\))
• Now, \(0.385<0.4<0.429\) is true as \(0.385 < 0.4\) and \(0.4<0.429\). But let's check other options to be sure.

Option B: \(\frac{6}{13}<\frac{3}{7}<\frac{2}{5}\)

• Convert to decimals:
• \(\frac{6}{13}\approx0.462\) (\(6\div13\approx0.462\))
• \(\frac{3}{7}\approx0.429\)
• \(\frac{2}{5}=0.4\)
• Here, \(0.462<0.429<0.4\) is false because \(0.462>0.429\) and \(0.429 > 0.4\).

Option C: \(\frac{2}{5}<\frac{3}{7}<\frac{6}{13}\)

• Convert to decimals:
• \(\frac{2}{5}=0.4\)
• \(\frac{3}{7}\approx0.429\)
• \(\frac{6}{13}\approx0.462\)
• The order here would be \(0.4 < 0.429<0.462\), but the option says \(\frac{2}{5}<\frac{3}{7}<\frac{6}{13}\) which is correct in order? Wait, no, let's re - check. Wait, \(\frac{2}{5}=0.4\), \(\frac{3}{7}\approx0.429\), \(\frac{6}{13}\approx0.462\). So \(0.4 < 0.429<0.462\) is true in terms of order, but wait, earlier option A also seemed true? Wait, no, let's recalculate the values more accurately.

Wait, let's use common denominators. Let's find the least common multiple (LCM) of the denominators for each pair.

For \(\frac{2}{5}\) and \(\frac{3}{7}\):

• LCM of 5 and 7 is 35.
• \(\frac{2}{5}=\frac{2\times7}{5\times7}=\frac{14}{35}\), \(\frac{3}{7}=\frac{3\times5}{7\times5}=\frac{15}{35}\). So \(\frac{2}{5}<\frac{3}{7}\) (since \(14 < 15\)).

For \(\frac{5}{13}\) and \(\frac{2}{5}\):

• LCM of 13 and 5 is 65.
• \(\frac{5}{13}=\frac{5\times5}{13\times5}=\frac{25}{65}\), \(\frac{2}{5}=\frac{2\times13}{5\times13}=\frac{26}{65}\). So \(\frac{5}{13}<\frac{2}{5}\) (since \(25 < 26\)).

For \(\frac{3}{7}\) and \(\frac{6}{13}\):

• LCM of 7 and 13 is 91.
• \(\frac{3}{7}=\frac{3\times13}{7\times13}=\frac{39}{91}\), \(\frac{6}{13}=\frac{6\times7}{13\times7}=\frac{42}{91}\). So \(\frac{3}{7}<\frac{6}{13}\) (since \(39 < 42\)).

Now, let's re - evaluate option A: \(\frac{5}{13}<\frac{2}{5}<\frac{3}{7}\)
We have \(\frac{5}{13}<\frac{2}{5}\) (from above, \(\frac{5}{13}=\frac{25}{65}\), \(\frac{2}{5}=\frac{26}{65}\)) and \(\frac{2}{5}<\frac{3}{7}\) (from above, \(\frac{2}{5}=\frac{14}{35}\), \(\frac{3}{7}=\frac{15}{35}\)). So \(\frac{5}{13}<\frac{2}{5}<\frac{3}{7}\) is true? Wait, but \(\frac{3}{7}\approx0.429\) and \(\frac{6}{13}\approx0.462\), so \(\frac{3}{7}<\frac{6}{13}\). So option A's last inequality \(\frac{2}{5}<\frac{3}{7}\) is true, and \(\frac{5}{13}<\frac{2}{5}\) is true. But let's check option D:

Option D: \(\frac{2}{5}<\frac{5}{13}<\frac{3}{7}\)

• From earlier, \(\frac{5}{13}=\frac{25}{65}\), \(\frac{2}{5}=\frac{26}{65}\). So \(\frac{25}{65}<\frac{26}{65}\) means \(\frac{5}{13}<\frac{2}{5}\), not \(\frac{2}{5}<\frac{5}{13}\). So option D is false.

Wait, maybe I made a mistake earlier. Let's re - check the decimal values:

• \(\frac{5}{13}\approx0.3846\)
• \(\frac{2}{5} = 0.4\)
• \(\frac{3}{7}\approx0.4286\)
• \(\frac{6}{13}\approx0.4615\)

So:

• Option A: \(0.3846<0.4<0.4286\) → True? Wait, \(0.4 < 0.4286\) is true, \(0.3846<0.4\) is true. So \(\frac{5}{13}<\frac{2}{5}<\frac{3}{7}\) is true.

• Option B: \(0.4615<0.4286<0.4\) → False (since \(0.4615>0.4286\) and \(0.4286 > 0.4\))

• Option C: \(0.4<0.4286<0.4615\) → This is true in terms of \(\frac{2}{5}<\frac{3}{7}<\frac{6}{13}\), but let's check the original option C: \(\frac{2}{5}<\frac{3}{7}<\frac{6}{13}\) → Yes, \(0.4<0.4286<0.4615\) is true. Wait, now I'm confused. Wait, the problem says "the fractions \(\frac{2}{5},\frac{5}{13},\) and \(\frac{3}{7}\)". Wait, maybe the options were misread. Let's check the original problem again. The user's second question: "Which of the following inequalities is true for the fractions \(\frac{2}{5},\frac{5}{13},\) and \(\frac{3}{7}\)?"

So the fractions are \(\frac{2}{5}\), \(\frac{5}{13}\), \(\frac{3}{7}\). So we need to compare these three.

Let's list their decimal equivalents:

• \(\frac{2}{5}=0.4\)

• \(\frac{5}{13}\approx0.3846\)

• \(\frac{3}{7}\approx0.4286\)

Now, let's order them:

\(0.3846<0.4<0.4286\) → \(\frac{5}{13}<\frac{2}{5}<\frac{3}{7}\) → which is option A.

So: