Question

independent practice leveled practice for 5-10, decompose mixed number in two different ways. us 5. \\(\frac{4}{6}=\\) \\(\frac{4}{6}=\\) 7. \\(1\frac{3}{5}=\\) \\(1\frac{3}{5}=\\) 9. \\(\frac{9}{12}=\\) \\(\frac{9}{12}=\\)

Explanation:

Problem 5: Decompose $\boldsymbol{\frac{4}{6}}$ in two different ways

Step 1: Decompose into unit fractions

A unit fraction has a numerator of 1. We can write $\frac{4}{6}$ as the sum of 4 unit fractions with denominator 6.
$\frac{4}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}$

Step 2: Decompose into two fractions

We can split the numerator 4 into two parts, say 2 and 2. Then we have:
$\frac{4}{6}=\frac{2}{6}+\frac{2}{6}$ (and we can simplify $\frac{2}{6}$ to $\frac{1}{3}$, so $\frac{4}{6}=\frac{1}{3}+\frac{1}{3}$)

Step 1: Decompose into a whole number and a fraction

A mixed number $1\frac{3}{5}$ can be written as the sum of the whole number part and the fractional part.
$1\frac{3}{5}=1+\frac{3}{5}$

Step 2: Decompose into fractions (improper fraction first, then split)

First, convert the mixed number to an improper fraction: $1\frac{3}{5}=\frac{1\times5 + 3}{5}=\frac{8}{5}$. Now, split the numerator 8 into two parts, say 5 and 3. Then:
$\frac{8}{5}=\frac{5}{5}+\frac{3}{5}=1+\frac{3}{5}$ (wait, that's the same as before. Let's split into other parts, like 4 and 4. $\frac{8}{5}=\frac{4}{5}+\frac{4}{5}$, but we need to relate it to the mixed number. Alternatively, $1\frac{3}{5}=\frac{5}{5}+\frac{3}{5}$ (which is the same as $1+\frac{3}{5}$) or we can write $1\frac{3}{5}=\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{3}{5}$ (but that's more unit fractions). Wait, better: $1\frac{3}{5}=\frac{6}{5}+\frac{2}{5}$ (since $1=\frac{5}{5}$, so $\frac{5}{5}+\frac{3}{5}=\frac{6}{5}+\frac{2}{5}$? Wait, no, $1\frac{3}{5}=\frac{8}{5}$, so $\frac{8}{5}=\frac{7}{5}+\frac{1}{5}=1\frac{2}{5}+\frac{1}{5}$ or $\frac{6}{5}+\frac{2}{5}=1\frac{1}{5}+\frac{2}{5}$ or $\frac{4}{5}+\frac{4}{5}= \frac{4}{5}+\frac{4}{5}$ (but $1\frac{3}{5}=\frac{8}{5}$, so $\frac{4}{5}+\frac{4}{5}=\frac{8}{5}$). Wait, maybe a better way: $1\frac{3}{5}= \frac{5}{5}+\frac{3}{5}$ (as $1+\frac{3}{5}$) and $1\frac{3}{5}=\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{3}{5}$ (but that's 1 + 3/5). Wait, another approach: $1\frac{3}{5}= \frac{8}{5}$, so we can write $\frac{8}{5}=\frac{5}{5}+\frac{3}{5}$ (which is $1+\frac{3}{5}$) and $\frac{8}{5}=\frac{4}{5}+\frac{4}{5}$ (since 4 + 4 = 8).

Step 1: Decompose into unit fractions

$\frac{9}{12}$ can be written as the sum of 9 unit fractions with denominator 12.
$\frac{9}{12}=\frac{1}{12}+\frac{1}{12}+\cdots+\frac{1}{12}$ (9 times)

Step 2: Decompose into simplified fractions

First, simplify $\frac{9}{12}=\frac{3}{4}$. Now, split $\frac{3}{4}$ into two fractions. For example, $\frac{3}{4}=\frac{1}{4}+\frac{2}{4}=\frac{1}{4}+\frac{1}{2}$. Also, we can split the original fraction: $\frac{9}{12}=\frac{6}{12}+\frac{3}{12}=\frac{1}{2}+\frac{1}{4}$ (since $\frac{6}{12}=\frac{1}{2}$ and $\frac{3}{12}=\frac{1}{4}$)

Answer:

$\frac{4}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}$; $\frac{4}{6}=\frac{1}{3}+\frac{1}{3}$ (or other valid decompositions like $\frac{3}{6}+\frac{1}{6}=\frac{1}{2}+\frac{1}{6}$)

Problem 7: Decompose $\boldsymbol{1\frac{3}{5}}$ in two different ways