Combining Rational Number Like Terms

Combining Rational Number Like Terms exercise

• Describe how to combine like terms, and simplify the expression $\frac{2}{3}h+\frac{1}{4}-\frac{1}{9}h+\frac{1}{4}$.

  Hints

  The commutative property of addition states that $a+b=b+a$.

  The commutative property of multiplication states that $a\times b=b\times a$.

  Reducing a fraction means dividing the numerator as well as the denominator by the same number.

  You can add terms such as $3h$ and $4h$ as follows: $3h+4h=(3+4)h=7h$.

  Solution

  To simplify $\frac23h+\frac14-\frac19h+\frac14$, you first reorder the terms using the commutative property of addition:

  $\frac23h-\frac19h+\frac14+\frac14$.

  Next, combine the like terms. For this you have to multiply $\frac23$ by $\frac33$. So both coefficients of $h$ have the same denominator:

  $\frac33\cdot\frac23h-\frac19h+\frac14+\frac14=\frac69h-\frac19h+\frac14+\frac14$.

  Now you are able to combine like terms:

  $\frac69h-\frac19h+\frac14+\frac14=\frac59h+\frac24=\frac59h+\frac12$.

  The second fraction $\frac24$ is reduced by dividing both the numerator and the denominator by $2$:

  $\frac24=\frac{2\div2}{4\div 2}=\frac12$.

• Explain the mathematical terms.

  Hints

  The commutative property for multiplication states that $a\times b=b\times a$.

  Expanding a fraction means multiplying the numerator as well as the denominator by the same number. This doesn't change the value of the fraction, as what you are multiplying by is equal to $1$.

  In this expression $3$ is the coefficient of $x^2$, $4$ is the coefficient of $x$, and $-4$ is a constant.

  Solution

  To combine rational number like terms we use some mathematical terms:

  • A constant is a number that is multiplied by a variable raised to the zeroth power.
  • A coefficient is a constant by which a variable is multiplied.
  • We use the commutative property of addition, $a+b=b+a$, to change the order of terms.
  • If we want to factor any term, like $a\times(b+c)$, we use the distributive property, which tells us that $a\times(b+c)=a\times b+a\times c$.
  • The multiplicative identity property of $1$ is that multiplying by it doesn't change the value.

• Give the steps for combing like terms.

  Hints

  The distributive property states that $a\times(b+c)=a\times b+a\times c$.

  The commutative property of addition states that a+b=b+a.

  For the standard form you reorder the terms from terms with the largest exponent to terms with the smallest exponent. For instance, $2x^2+2x-3y+4$ is in standard form.

  Solution

  First we rewrite the mixed fraction:

  $\frac{1}{3}+\frac{1}{2}\left(1\frac{1}{3}x-\frac{1}{4}\right)=\frac{1}{3}+\frac{1}{2}\left(\frac{4}{3}x-\frac{1}{4}\right)$.

  Next we use the distributive property to get rid of the parentheses:

  $\frac{1}{3}+\frac{1}{2}\left(\frac{4}{3}x-\frac{1}{4}\right)=\frac{1}{3}+\frac{4}{6}x-\frac{1}{8}$.

  Reordering the expression depending on the like terms leads to

  $\frac{1}{3}+\frac{4}{6}x-\frac{1}{8}=\frac{1}{3}-\frac18+\frac{4}{6}x$.

  So, now we are in the position to combine the like terms. For this we have to find the common denominator of the first two fractions, it's $24$, and expand the fraction as follows:

  $\begin{array}{rcl} \frac{1}{3}-\frac18+\frac{4}{6}x&=&\frac{1\cdot 8}{3\cdot 8}-\frac{1\cdot 3}{8\cdot 3}+\frac{4}{6}x\\ &=&\frac{8}{24}-\frac{3}{24}+\frac{4}{6}x\\ &=&\frac{5}{24}+\frac{4}{6}x \end{array}$

  Last we can reduce the second fraction and rewrite the expression in standard form:

  $\frac{5}{24}+\frac{4}{6}x=\frac{5}{24}+\frac23x=\frac23x+\frac{5}{24}$.

• Calculate the total amount of ingredients by combining like terms.

  Hints

  Let $e$ be the number of eggs, $f$ the number of cups of flour, and $s$ the number of cups of sugar.

  For the muffins, Laura needs $2\frac13f+\frac12e$.

  For the cookies, Laura needs $\frac12\left(1\frac13s+\frac34f\right)+\frac14e$.

  Just add like terms and combine.

  Solution

  First we establish both expressions describing the ingredients for the muffins as well as for the cookies:

  • muffins: $2\frac13f+\frac12e$
  • cookies: $\frac12\left(1\frac13s+\frac34f\right)+\frac14e$

  where $e$ is the number of eggs, $f$ is the number of cups of flour, and $s$ the number of cups of sugar.

  We add both expressions together to get,

  $2\frac13f+\frac12e+\frac12\left(1\frac13s+\frac34f\right)+\frac14e$.

  Rewrite the mixed terms as fractions and use the distributive property:

  $\frac73f+\frac12e+\frac12\left(\frac43s+\frac34f\right)+\frac14e=\frac73f+\frac12e+\frac46s+\frac38f+\frac14e$.

  Now we reorder the expression to get,

  $\frac73f+\frac38f+\frac12e+\frac14e+\frac46s$.

  Almost done! We still have to find the common denominators to combine like terms:

  $\begin{array}{rcl}\frac73f+\frac38f+\frac12e+\frac14e+\frac46s&=&\frac{7\cdot 8}{3\cdot 8}f+\frac{3\cdot 3}{8\cdot 3}f+\frac{1\cdot2}{2\cdot 2}e+\frac14e+\frac46s\\ &=&\frac{56}{24}f+\frac{9}{24}f+\frac{2}{4}e+\frac14e+\frac46s\\ &=&\frac{65}{24}f+\frac34e+\frac23s \end{array}$

  So in total Laura needs:

  • $\frac34$ eggs
  • $\frac{65}{24}$ cups of flour
  • $\frac23$ cups of sugar

• State how to combine like terms with integers.

  Hints

  The commutative property of addition is given by $a+b=b+a$.

  Just look at the following examples for like terms:

  • $4x$ and $-3x$ are like terms.
  • $4x$ and $4y$ aren't like terms at all.
  • $3y$ and $-4y$ are like terms.

  Solution

  To simplify this expression we first find the like terms and reorder the expression using the commutative property.

  So the $x$-terms as well as the $y$-terms are grouped together: $2x+13x-7y+9y$.

  Now we can combine the like terms:

  • $2x+13x=15x$
  • $-7y+9y=2y$

  This gives us the simplified expression $15x+2y$.

• Combine and simplify the given rational expressions as much as possible.

  Hints

  Use the commutative property, $a+b=b+a$, as well as the distributive property, $a\times(b+c)=a\times b+a\times c$.

  Here you see an example for changing mixed terms into fractions.

  Solution

  • $\mathbf{\frac23\left(\frac12x-\frac14+x\right)}$

  With the distributive property we get,

  $\frac23\left(\frac12x-\frac14+x\right)=\frac26x-\frac2{12}+\frac23x$.

  We can now reduce the fractions and reorder the expression to get

  $\frac13x+\frac23x-\frac16=x-\frac16$.

  $~$

  • $\mathbf{\frac45x-\frac13y+\frac23y-y+\frac15x}$

  In this example, we reorder the expression to get

  $\frac45x+\frac15x-\frac13y+\frac23y-y$.

  Now we can combine the like terms to get

  $\frac55x+\frac13y-y=x+\frac13y-\frac33y=x-\frac23y$.

  $~$

  • $\mathbf{3\left(\frac25y-\frac23x+2x\right)-\frac15y}$

  Again we use the distributive property to get rid of the parentheses:

  $\frac65y-\frac63x+6x-\frac15y$.

  Now we reduce the fractions if possible, reorder the expression, and combine the like terms:

  $-2x+6x+\frac65y-\frac15y=4x+\frac55y=4x-y$.

  $~$

  • $\mathbf{\frac38y+\frac25x+\frac56y-\frac4{15}x}$

  Here we have to reorder the expression and find the common denominator:

  $\begin{array}{rcl} \frac38y+\frac25x+\frac56y-\frac4{15}x&=&\frac25x-\frac4{15}x+\frac38y+\frac56y\\ &=&\frac6{15}x-\frac4{15}x+\frac9{24}y+\frac{20}{24}y\\ &=&\frac2{15}x+\frac{29}{24}y \end{array}$

  $~$

  • $\mathbf{2\frac12\left(\frac15x-\frac35y+\frac32x+y\right)}$

  First we look at the term inside the parentheses:

  $\frac15x-\frac35y+\frac32x+y=\frac2{10}x+\frac{15}{10}x-\frac35y+\frac55y=\frac{17}{10}x+\frac25y$.

  The mixed number $2\frac12$ can be written as a fraction, $\frac52$. Last we use the distributive property:

  $\frac52\left(\frac{17}{10}x+\frac25y\right)=\frac{85}{20}x+\frac{10}{10}y=\frac{17}4x+y$.

  $~$

  • $\mathbf{3\frac13x+2\frac14y-2\frac12x-1\frac15y}$

  First of all we write each mixed number as a fraction:

  $\frac{10}3x+\frac94y-\frac52x-\frac65y$.

  Now we reorder the expression to

  $\frac{10}3x-\frac52x+\frac94y-\frac65y$.

  Next we expand each fraction to get the common denominator:

  $\frac{20}{6}x-\frac{15}6x+\frac{45}{20}y-\frac{24}{20}y=\frac56x+\frac{21}{20}y$.