Understanding Equations

Understanding Equations exercise

• Determine the number of days Phineas Fogg will travel by steamer.

  Hints

  Subtract the number of days which Phineas travels by train from the total number of the days of his trip.

  We have $23-(4+12)=23-16=7$, or $23-(4+12)=23-4-12=19-12=7$.

  Solution

  To determine the time Phineas Fogg has to travel by steamboat, we must subtract the number of the days Phineas travels by train from $79$ days.

  He travels by the train during five legs of his trip. Two legs take $13$ days, or $2\cdot 13$ days. Two legs take $7$ days, or $2\cdot 7$ days. And one leg takes $22$ days. That is a total of $2\cdot 13+2\cdot7+22=62$ day by train.

  Now we subtract this number from $79$ to get $79-62=17$ days that he will travel by steamboat.

• Examine the costs of Phineas Fogg's trip.

  Hints

  Adding a value to itself twice is the same as multiplying it by 2: $x+x=2\cdot x$.

  Similarly, Adding a value to itself three times is the same as multiplying it by 3: $x+x+x=3\cdot x$.

  You have to add the price of the steamboat tickets and the train tickets together.

  Pay attention to PEMDAS.

  Solution

  Phineas needs three steamboat tickets for the price of $S$, and five train tickets for the price of $T$. So the cost $C$ of the all of his tickets is $C=3\times S+5\times T$.

  The relation between the price of the steamboat tickets and the price of the train ticket is given by $T=\frac23\times S$.

  This means that $C=3\times S+5\times T=3\times S+5\times (\frac23\times S)$.

  Plug in the known price for one steamboat ticket:

  $\begin{array}{rcl} C&=&3\cdot 45+5\cdot \frac23 \cdot 45\\ &=&135+150\\ &=&285 \end{array}$

  So we can see that Phineas needs to pay a total of $285$ pounds on tickets for his trip.

• Identify the solutions of the equations.

  Hints

  The keywords each or for one indicate multiplication.

  Take a look at the following example:

  If $1$ liter of milk costs $1.20$ pounds, then $5$ liters cost $5\times1.20=6$ pounds.

  If you buy, in addition, $6$ bread rolls at $0.50$ pounds each, you have to pay

  $5\times 1.20+6\times 0.50=6+3=9$ pounds altogether.

  Remember PEMDAS: multiplication comes before addition.

  Solution

  Coffee and muffin

  • A coffee costs $2.50$ pounds. So we get $4\times 2.50$ pounds for $4$ coffees.
  • A muffin costs $2$ pounds. This leads to $3\times2$ pounds for $3$ muffins.
  • To receive the whole price, these expression have to be added: $4\times 2.50+3\times 2=10+6=16$.

  The total costs for the coffees and muffins are $16$ pounds.

  $~$

  Tickets and ice

  • A ticket costs $4$ pounds. So $3$ tickets $3\times 4$ pounds.
  • An ice costs $1.50$ pounds and hence $4$ ices $4\times1.50$ pounds.
  • These Terme are added to reach the whole price: $3\times 4+4\times1.50=12+6=18$.

  $18$ pounds must be given away for the tickets and the ice.

  $~$

  Taco and coke

  • A taco costs $4.50$ pounds, so $2$ tacos cost $2\times 4.50$ pounds.
  • If a coke costs $2.50$ pounds, $4$ cokes cost $4\times 2.50$ pounds.
  • Now these Terme must be still added: $2\times 4.50+4\times2.50=9+10=19$.

  Tacos and coke cost $19$ pounds together.

  $~$

  Bagel and tea

  • A Bagel costs $1.20$ pounds and with it $5$ Bagel $5\times 1.20$ pounds.
  • A tea costs $3$ pounds, then $4$ teas $4\times3$ pounds.
  • Now these Terme can be added: $5\times 1.20+4\times 3=6+12=18$.

  Bagel and tea cost $19$ pounds in total.

• Examine the equation and solve it.

  Hints

  Phineas drinks $2$ tea each day.

  You have to multiply the price of each food by the number of days.

  Phineas spends a little more than $50$ pounds. So he has a little less than $50$ pounds at disposal.

  Solution

  First we calculate the total costs for the mango sticky rice and Thai iced tea separately:

  • Mango sticky rice: $30\times 0.75=22.50$
  • Thai ice tea: $30\times 2\times 0.50=30$

  Adding these costs together, we see that Phineas spends $22,50+30=52.50$ pounds on mango sticky rice and Thai ice tea during his holidays.

  He would like to know how much money he has at his disposal to spend on other foods. To figure this out, he subtracts $52.50$ from $100$ pounds: $100-52.50=47.50$.

  So we see that Phineas has $47.50$ pounds to spend on other foods.

• Decide if the statement is true or false.

  Hints

  An expression is a meaningful arrangement of numbers, variables, and operation signs.

  A number is an expression.

  The addition, subtraction, multiplication, or division of expressions is an expression too.

  In an equation, there are expressions on the left-hand side as well as the right-hand side of an equals sign.

  Solution

  In general an expression is a meaningful arrangement of numbers, variables, and operations, as well as parentheses.

  An equation contains an equals sign.

  In an equation, there are expressions on the left-hand side as well as the right-hand side of an equals sign.

  A single number is an expression.

  Here are some examples of expressions:

  • $2\cdot 13+2\cdot 7+22$
  • $79-62$

  These expressions are not equations because there are no equals signs.

  An example of an equation is $3\cdot 45+5\cdot \frac23\cdot 45=285$.

• Find the solution to the word problem.

  Hints

  The first monkey throws $5$ balls to the second one. Then the elephant comes into play; he takes a ball.

  After $5$ balls have been thrown, the elephant takes one. Then the juggling $4$ balls before the first monkey throws him a new ball.

  After the first $5$ balls, the elephant is always taking $1$ ball from the second monkey after the first monkey throws him a ball.

  Solution

  How can we find the number of balls the elephant throws into the net if the first monkey has $25$ balls to throw? Let's see what is going on step-by-step:

  • The first monkey throws the first ball to the second monkey. Now the second monkey is juggling $1$ ball, and the first monkey has $24$ balls left to throw.
  • The first monkey throws the second ball to the second monkey. Now the second monkey is juggling $2$ balls, and the first monkey has $23$ balls left to throw.
  • The first monkey throws the third ball to the second monkey. Now the second monkey is juggling $3$ balls, and the first monkey has $22$ balls left to throw.
  • The first monkey throws the fourth ball to the second monkey. Now the second monkey is juggling $4$ balls, and the first monkey has $21$ balls left to throw.
  • The first monkey throws the fifth ball to the second monkey. Now the second monkey is juggling $5$ balls, and the first monkey has $20$ balls left to throw.

  Since the second monkey is juggling $5$ balls, the elephant takes one of these balls and throws it into the net. Now there is one ball in the net and second monkey is juggling $4$ balls again.

  Now, every time the first monkey throws the second monkey a ball, the elephant will take a ball and throw it into the next. Since there are $20$ balls left to throw, the elephant will do this $20$ times. And so that is $20$ balls in the net plus $1$ ball from the initial $5$ balls. All together, there are $21$ balls in the net by the time the animals stop and take a bow.

  What about when the first monkey has $b$ balls to throw?

  Well, we have the initial number of balls $b$ the first monkey has to throw. Then, after the first four balls are thrown, a ball gets thrown into the net every time the first monkey throws a ball to the second monkey. After all $b$ balls have been thrown, there are 4 left over which have not been thrown into the net. So every ball but four end up getting thrown into the net, meaning that $b-4$ balls end up in the net in the end.