Question

subtract. simplify, if possible.
$4\frac{2}{7} - 1\frac{9}{14}$
? $\frac{\square}{\square}$
enter the whole number.

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert \(4\frac{2}{7}\) to an improper fraction. The formula for converting a mixed number \(a\frac{b}{c}\) to an improper fraction is \(\frac{a\times c + b}{c}\). So for \(4\frac{2}{7}\), we have \(a = 4\), \(b = 2\), \(c = 7\). Then \(\frac{4\times7 + 2}{7}=\frac{28 + 2}{7}=\frac{30}{7}\).

For \(1\frac{9}{14}\), using the same formula with \(a = 1\), \(b = 9\), \(c = 14\), we get \(\frac{1\times14+9}{14}=\frac{14 + 9}{14}=\frac{23}{14}\).

Step2: Find a common denominator

The denominators are 7 and 14. The least common denominator of 7 and 14 is 14. So we convert \(\frac{30}{7}\) to a fraction with denominator 14. To do this, we multiply the numerator and denominator by 2: \(\frac{30\times2}{7\times2}=\frac{60}{14}\).

Step3: Subtract the fractions

Now we subtract \(\frac{23}{14}\) from \(\frac{60}{14}\). When subtracting fractions with the same denominator, we subtract the numerators and keep the denominator: \(\frac{60 - 23}{14}=\frac{37}{14}\).

Step4: Convert back to a mixed number

Now we convert \(\frac{37}{14}\) back to a mixed number. We divide 37 by 14: \(14\times2 = 28\), \(37-28 = 9\). So \(\frac{37}{14}=2\frac{9}{14}\)? Wait, no, wait. Wait, original mixed numbers: \(4\frac{2}{7}\) is \(4 + \frac{2}{7}\), \(1\frac{9}{14}\) is \(1+\frac{9}{14}\). Maybe a better way is to handle the whole numbers and fractions separately.

Wait, let's try again. \(4\frac{2}{7}=4+\frac{2}{7}\), \(1\frac{9}{14}=1+\frac{9}{14}\). Let's convert \(\frac{2}{7}\) to fourteenths: \(\frac{2}{7}=\frac{4}{14}\). So \(4\frac{2}{7}=4+\frac{4}{14}=3 + 1+\frac{4}{14}=3+\frac{14 + 4}{14}=3\frac{18}{14}\). Now subtract \(1\frac{9}{14}\) from \(3\frac{18}{14}\). Subtract the whole numbers: \(3 - 1 = 2\). Subtract the fractions: \(\frac{18}{14}-\frac{9}{14}=\frac{9}{14}\). So the result is \(2\frac{9}{14}\)? Wait, no, wait, \(4\frac{2}{7}\) is \(4 + \frac{2}{7}\), if we borrow 1 from the whole number 4, we get \(3 + 1+\frac{2}{7}=3+\frac{7 + 2}{7}=3\frac{9}{7}\)? No, that's not right. Wait, \(\frac{2}{7}=\frac{4}{14}\), so \(4\frac{2}{7}=4\frac{4}{14}\). Now \(4\frac{4}{14}-1\frac{9}{14}\). Since \(\frac{4}{14}<\frac{9}{14}\), we need to borrow 1 from the whole number 4, making it \(3\), and add \(\frac{14}{14}\) to the fraction: \(3 + \frac{14 + 4}{14}=3\frac{18}{14}\). Now subtract \(1\frac{9}{14}\): \(3 - 1 = 2\), \(\frac{18}{14}-\frac{9}{14}=\frac{9}{14}\). So the result is \(2\frac{9}{14}\)? Wait, but that seems low. Wait, \(4\frac{2}{7}\) is approximately 4.2857, \(1\frac{9}{14}\) is approximately 1.6429. Subtracting them: 4.2857 - 1.6429 = 2.6429, and \(2\frac{9}{14}\) is \(2 + \frac{9}{14}\approx2 + 0.6429 = 2.6429\), which matches. Wait, but the problem says "Enter the whole number" and then the fraction. Wait, maybe I made a mistake earlier. Wait, let's check the steps again.

Wait, \(4\frac{2}{7}\) is equal to \(\frac{4\times7 + 2}{7}=\frac{30}{7}\). \(1\frac{9}{14}=\frac{1\times14 + 9}{14}=\frac{23}{14}\). Now, \(\frac{30}{7}=\frac{60}{14}\). Then \(\frac{60}{14}-\frac{23}{14}=\frac{37}{14}\). Now, \(\frac{37}{14}\) as a mixed number: 14*2=28, 37-28=9, so \(2\frac{9}{14}\). So the whole number is 2, and the fraction is \(\frac{9}{14}\). Wait, but the problem's box is for the whole number, then the fraction. So the whole number is 2? Wait, no, wait, 4 - 1 is 3, but the fraction part: \(\frac{2}{7}-\frac{9}{14}\). Since \(\frac{2}{7}=\frac{4}{14}\), which is less than \(\frac{9}{14}\), so we need to borrow 1 from the whole number 4, making it 3, and add \(\frac{14}{14}\) to \(\frac{4}{14}\), so we have \(3 + \frac{18}{14}\), then subtract \(1\frac{9}{14}\): 3 - 1 = 2, \(\frac{18}{14}-\frac{9}{14}=\frac{9}{14}\), so the result is \(2\frac{9}{14}\). So the whole number is 2.

Wait, but let's verify with decimals: \(4\frac{2}{7}\approx4.2857\), \(1\frac{9}{14}\approx1.6429\), subtraction: 4.2857 - 1.6429 = 2.6429, which is \(2\frac{9}{14}\) (since \(\frac{9}{14}\approx0.6429\)). So the whole number is 2.

Answer:

The whole number is \(\boxed{2}\), and the fraction is \(\frac{9}{14}\) (but the question asks for the whole number, so the answer for the whole number is 2).