Keywords for Division

Keywords for Division exercise

• Describe different ways to represent division.

  Hints

  Check the result by inverting division.

  $?\cdot 6=18$

  Keep in mind:

  dividend $\div$ divisor $=$ quotient.

  You have different ways to write down a division, but it doesn't make sense to get different results.

  Solution

  The dog has to divide the flock of $18$ sheep into groups of $6$.

  This is a division task: $18\div 6$.

  • $18$ is the dividend, the number you are going to divide.
  • $6$ is the divisor, the number of sheep in each group.
  • The result, the number of groups of sheep, is the quotient. Here it is $3$.

  You can write division in different ways:

  • divisor)$\overline{\text{dividend}}$ and for the example above $6)\overline{18}$
  • dividend $\div$ divisor - this leads to $18\div 6$
  • $\frac{\text{dividend}}{\text{divisor}}$ - or for the example above $\frac{18}6$

  No matter which way of writing down division you use, the result is always the same. In this case, this is the quotient $3$.

• Explain ratios.

  Hints

  Keep in mind that we are looking for the ratio of white sheep to black sheep. Not the other way round.

  You can reduce the fraction as follows:

  $\frac{15}3=\frac51=5$.

  But a ratio needs two arguments.

  Solution

  You can also describe ratios using division notation.

  If you look at the example of $18$ sheep with...

  • ...$15$ white sheep and
  • ...$3$ black sheep

  you get the ratio of "white sheep" to "black sheep" by dividing $15$ by $3$.

  So $15:3$ is the ratio of white sheep $:$ black sheep.

  You can simplify the expression to $5:1$. This means that for each group of $5$ white sheep, there is just $1$ black sheep in the group.

  You can also write a ratio...

  • ...as a fraction $\frac51$, or
  • ...as $5$ to $1$.

• Describe how to determine the average.

  Hints

  Let's have a look at another average calculation, for George and Ringo:

  • First, we add the given pounds to get $4+6=10$.
  • Next, we divide this sum by the number of sheep, i.e. $2$.

  This gives us an average of $10\div 2=5$.

  The sum of all calculated averages above is $11$.

  Two of the averages are the same.

  Solution

  To determine the average of different values you proceed as follows:

  1. Add all the given values.
  2. Divide the resulting sum by the number of values.

  Let's start with John and Paul:

  1. $4+2=6$
  2. $6\div 2=3$

  So the average number of pounds of wool given by John and Paul is $3$.

  Next we consider John, Paul, and Ringo:

  1. $2+4+6=12$
  2. $12\div3=4$

  The average wool production of these three sheep is given by $4$ pounds.

  Lastly, let's have a look at all the sheep:

  1. $4+2+4+6=16$
  2. $16\div4=4$

  All the sheep together produce $4$ pounds of wool on average.

• Highlight the key words concerning division problems.

  Hints

  Remember, dividend $\div$ divisor $=$ quotient.

  Here you see an example with a remainder:

  $25\div 3=7$ remainder $4$.

  Let's check it: $7\times 3=21$. So you get the following:

  $7\times 3+4=21+4=25$.

  Solution

  Remember, dividend $\div$ divisor $=$ quotient:

  Dividing stands for...

  • ...splitting into several parts
  • ...sharing anything
  • ...cutting into equal pieces

  It also can be used for ratios or averages.

  Paul shares his cherry sweets leads to $25\div 5=5$:

  • The number of sweets, $25$, is the dividend.
  • The number of persons, $5$, is the divisor.
  • The resulting number of sweets for each person, $5$, is the quotient.

  Anne wants to do all of her homework assignments in $3$ days:

  • $12$ is the dividend.
  • $3$ is the divisor.
  • $12\div 3=4$ is the quotient.

  So she has to do $4$ homework assignments each day.

  The average age of Claire, Sue, and Jenny can be calculated as follows:

  • $17+21+34=72$. This is the dividend.
  • Divide the dividend by $3$, the number of people. This is the divisor.
  • The average age is the quotient $72\div 3=24$.

  In the last example we can examine a remainder: Monica wants to split the dividend $100$ dollars among $6$, the divisor, people.

  This leads to $100\div 6=16$ remainder $4$ dollars per person.

• Represent dividend, divisor, and quotient.

  Hints

  For addition, you have: (summand) $+$ (summand) $=$ (sum).

  For subtraction, you have: (minuend) $-$ (subtrahend) $=$ (difference).

  For multiplication, you have: (factor) $\times$ (factor) $=$ (product).

  The divisor is that which you divide the dividend.

  The quotient is the result of a division.

  Solution

  The dog has to divide the herd into groups of $6$. $18$ sheep belong to the herd. So you get the following division task: $18\div 6$.

  • $18$ is the dividend, the number you are going to divide.
  • $6$ is the divisor, the number of groups you are going to make.
  • The result, the number of groups of $6$ sheep made, is the quotient. Here it is $3$.

  So you can remember the following: dividend $\div$ divisor $=$ quotient.

• Apply division, ratio, and average problems.

  Hints

  Simplify ratios or fractions if it's possible.

  Let's have a look at the following example:

  $\frac{20}{8}=\frac 52$.

  To get the average of several values proceed as follows:

  • Add the values.
  • Divide the resulting sum by the number of values.

  For the desired ratio, determine the number of female students first.

  Solution

  Let's start with the ratio male students to female students:

  • There are $30$ students in total, $18$ of them male.
  • Thus $30-18=12$ are female students.

  This leads to the ratio $18$ to $12$, which can simplified to $3$ to $2$.

  You can interpret this result as follows: for each $3$ male students there are $2$ female students.

  To practice division the school class will be divided into four groups. $30$ isn't divisible by $4$. Thus we must get a remainder. We have to divide $30$ by $4$ as follows:

  $30\div 4=7$ remainder $2$.

  Perhaps those $2$ students can be the referees for the following quiz. We have to determine the average:

  1. Add all given values: $120+80+70=270$.
  2. Divide this sum by the number of values $270\div 3=90$.

  The desired average score is $90$.