Question

1. determine a number between the two given numbers.

a) \\(\frac{4}{3}\\) __ \\(\frac{7}{6}\\)
b) \\(2\frac{1}{10}\\) __ \\(\frac{11}{5}\\)
c) \\(5\frac{1}{5}\\) __ 5.3
d) 1.3 __ \\(\frac{10}{6}\\)

Explanation:

Part a) Step1: Find a common denominator

The denominators are 3 and 6, so the common denominator is 6. Convert $\frac{4}{3}$: $\frac{4\times2}{3\times2}=\frac{8}{6}$. Now we have $\frac{8}{6}$ and $\frac{7}{6}$. Wait, no, wait—wait, $\frac{4}{3}$ is approximately 1.333 and $\frac{7}{6}$ is approximately 1.166? Wait, no, I made a mistake. Wait, $\frac{4}{3} \approx 1.333$, $\frac{7}{6} \approx 1.166$? No, that can't be. Wait, no, $\frac{7}{6}$ is about 1.166, $\frac{4}{3}$ is about 1.333. Wait, but we need a number between them. Wait, maybe find the average. The average of $\frac{4}{3}$ and $\frac{7}{6}$ is $\frac{\frac{4}{3}+\frac{7}{6}}{2}=\frac{\frac{8 + 7}{6}}{2}=\frac{\frac{15}{6}}{2}=\frac{15}{12}=\frac{5}{4}=1.25$. Let's check: $\frac{7}{6}\approx1.166$, $\frac{5}{4}=1.25$, $\frac{4}{3}\approx1.333$. Yes, that works. Alternatively, convert to decimals: $\frac{4}{3}\approx1.333$, $\frac{7}{6}\approx1.166$. A number between them could be 1.2, which is $\frac{6}{5}$, or 1.25 as above. Let's use the average method.

Part a) Step2: Calculate the average

Average of $\frac{4}{3}$ and $\frac{7}{6}$: $\frac{\frac{4}{3}+\frac{7}{6}}{2}=\frac{\frac{8 + 7}{6}}{2}=\frac{15}{12}=\frac{5}{4}=1.25$. So $\frac{5}{4}$ (or 1.25) is between them.

Part b) Step1: Convert mixed number to improper fraction

$2\frac{1}{10}=\frac{2\times10 + 1}{10}=\frac{21}{10}=2.1$. $\frac{11}{5}=2.2$. Wait, $2\frac{1}{10}=2.1$, $\frac{11}{5}=2.2$. So a number between them could be 2.15, which is $\frac{43}{20}$, or the average: $\frac{2.1 + 2.2}{2}=2.15$. Let's do it with fractions. $2\frac{1}{10}=\frac{21}{10}$, $\frac{11}{5}=\frac{22}{10}$. So a number between $\frac{21}{10}$ and $\frac{22}{10}$ is $\frac{21.5}{10}=\frac{43}{20}=2.15$.

Part b) Step2: Find the average

Average of $2\frac{1}{10}$ (2.1) and $\frac{11}{5}$ (2.2) is $\frac{2.1 + 2.2}{2}=2.15=\frac{43}{20}$.

Part c) Step1: Convert mixed number to decimal

$5\frac{1}{5}=\frac{5\times5 + 1}{5}=\frac{26}{5}=5.2$. We need a number between 5.2 and 5.3. So 5.25 (which is $\frac{21}{4}$ or 5.25) works.

Part c) Step2: Find a number between 5.2 and 5.3

5.25 is between 5.2 and 5.3.

Part d) Step1: Convert fraction to decimal

$\frac{10}{6}=\frac{5}{3}\approx1.666...$. We need a number between 1.3 and 1.666... So 1.4 (which is $\frac{7}{5}$) or 1.5 (which is $\frac{3}{2}$) works. Let's check the average: $\frac{1.3+\frac{10}{6}}{2}=\frac{1.3 + 1.666...}{2}=\frac{2.966...}{2}\approx1.483...$, which is also between them. Let's use 1.5 (3/2) as an example.

Part d) Step2: Find a number between 1.3 and $\frac{10}{6}$

$\frac{10}{6}\approx1.666$, 1.5 is between 1.3 and 1.666.

Answer:

s (Examples, as there are multiple correct answers):
a) $\boldsymbol{\frac{5}{4}}$ (or 1.25, or other numbers between $\frac{7}{6}$ and $\frac{4}{3}$)
b) $\boldsymbol{\frac{43}{20}}$ (or 2.15, or other numbers between $2\frac{1}{10}$ and $\frac{11}{5}$)
c) $\boldsymbol{5.25}$ (or $\frac{21}{4}$, or other numbers between $5\frac{1}{5}$ and 5.3)
d) $\boldsymbol{\frac{3}{2}}$ (or 1.5, or other numbers between 1.3 and $\frac{10}{6}$)