Clever Calculations with Money

Clever Calculations with Money exercise

• Explain how to use mental math using quarters.

  Hints

  If one drinks costs $ \$ 1.25$ dollars, five drinks costs

  $1.25+1.25+1.25+1.25+1.25$.

  You can do this more quickly by using multiplication.

  One quarter is worth $0.25$ dollars.

  Solution

  What a disaster: Blake and Denise can't recognize the total for their five drinks, which cost $ \$ 1.25 $ each.

  First, the keyword each indicates multiplication. So they have to multiply $5\times 1.25$.

  For this they first multiply $5\times 1=5$. That's quite easy.

  They still have to multiply $5\times 0.25$. Blake remembers that four quarters are in one dollar. So, there is still one left. This leads to

  $5\times 0.25=1.25$.

  Finally, they add the results $5+1.25=6.25$ to get the total for all their drinks: $6.25$ dollars.

• Explain grouping mental math strategies for addition and subtraction.

  Hints

  Here you see another example:

  $22+19=21+1+19=21+20=41$.

  A similar trick also works for subtraction:

  $16-8=16+2-2-8=18-10=8$.

  Solution

  Mental math is sometimes made easier with the help of a few tricks.

  Simplifying addition with compensating

  You adjust one number to have zero in the ones place by adding, and then subtract the same amount from the second number:

  • $19+26=19+1-1+26=20+25=45$.
  • $99+172=99+1-1+172=100+171=271$.

  This trick works for subtraction too:

  • $15-9=15+1-1-9=16-10=6$.

• Explain how to group division problems.

  Hints

  Divide step by step. Just look at the example $13.20 \div 2$:

  • $10\div 2= 5$
  • $3\div 2=0.75$
  • $0.20\div 2=0.10$

  Add all the results $5+0.75+0.10=5.85$

  Sometimes dividing leads to a remainder:

  $10\div 3= 3$ with the remainder $1$

  because

  $3\times 3+1=10$

  Solution

  Sometimes it's easier to divide step by step, and to add the individual results after:

  $\mathbf{45.20\div 2}$

  • $40\div2=20$
  • $5\div 2=2.50$
  • $0.20\div 2=0.10$

  Now we add the results: $20+2.50+0.10=22.60$

  The next example shows how to handle a remainder:

  $\mathbf{37.50\div 3}$

  • $30\div 3=10$
  • $7\div 3=2$ with a remainder of $1$
  • Add the remainder to the decimal and divide the sum by $3$: $1.50\div 3=0.50$

  Adding the results leads to $10+2+0.50=12.50$

  $\mathbf{22.80\div 4}$

  • $20\div 4=5$
  • $2\div 4=0.50$
  • $0.80\div 4=0.20$

  Once again, we get the final result by adding: $5+0.50+0.20=5.70$

• Solve the following problems involving money.

  Hints

  One quarter is $25$ cents, a dime is $10$ cents, and a nickel is $5$ cents.

  First decide if you have to add, subtract, multiply, or divide.

  Solution

  Cameron

  $0.25$ for each of the $8$ hats leads to multiplication: $8\times 0.25$. As $8$ is double $4$, it's easier to multiply a quarter by $4$ to get $1$ dollar, and to double the result to get $2$ dollars.

  $~$

  Sandy

  Just add the values: $6+2\times 0.25+5\times 0.10+8\times 0.05=6+0.50+0.50+0.40=6+1+0.40=7.40$ dollars.

  $~$

  Anne

  $5$ tacos at $3.40$ dollars each costs $5\times 3.40$:

  • $5\times 3=15$
  • $5\times 0.40=2$

  Adding these two results leads to the total of $17$ dollars.

  $~$

  Sakamma

  Here we have to divide: We know the total $33.50$ for $5$ burgers.

  • $30\div 5=6$
  • $3\div 5=0.60$
  • $0.50\div 5=0.10$

  Last, we add the results: $6+0.60+0.10=6.70$ dollars, the price for one burger.

• Determine the amount of money needed to pay each bill.

  Hints

  Count the number of the notes as well as the coins.

  Multiply the value of any bill or coin by the number of coins or bills of that type. Then add the resulting values.

  Solution

  Let's check the receipts from left to the right:

  The receipt for $45.20$ dollars

  Two $20$ dollar bills and one $5$ dollar bill, plus two $10$ cent coins.

  The receipt with the amount $25.20$ dollars

  Two $10$ dollar bills and one $5$ dollar bill, plus one $50$ cent piece and three $10$ cent coins.

  The receipt with the amount $38.50$ dollars

  One $20$ dollar bill, one $10$ dollar bill, one $5$ dollar bill, and three $1$ dollar bills, as well as one $50$ cent coin.

  The receipt with the amount $27.60$ dollars

  Two $10$ dollar bills, one $5$ dollar bill, and two $1$ dollar bill, plus one $50$ cent piece and one $10$ cent coin.

• Identify correct mental math techniques.

  Hints

  Just look at the following example $57\times 11$:

  • The last digit is $7$.
  • Add $5+7=12$.
  • The middle digit is $2$.
  • The first digit is $5+1=6$.

  So we get $57\times 11=627$.

  Check the statements with a few examples.

  Solution

  Let's check the tricks:

  Multiplying a whole number by $10$

  This leads to the same number with a $0$ tacked to the end:

  • $12\times 10=120$
  • $123\times 10 =1230$
  • ...

  After all, if you multiply a decimal by ten, you have to shift the decimal point to the right:

  • $1.2\times 10=12.0=12$
  • $1.23\times 10=12.3$

  $~$

  Multiplying by $9$

  $x\times 9=x\times (10-1)=10x-x$

  using the distributive property. Let's have a look at an example:

  $27\times 9=270-27=270-20-7=250-7=243$

  Another example for simplifying multiplying by $5$. As $5=10\div 2$ we can do the following:

  $x\times 5=x\div 2\times 10=x\times 10\div2$

  Again, we try an example: $120\times 5=60\times 10=600$

  You can always decide which calculation seems to be more comfortable for you.

  $~$

  Multiplying by $11$

  $49\times 11$:

  • The last digit is $9$.
  • Add $4+9=13$.
  • The middle digit is $3$.
  • The first digit is $4+1=5$.

  So we get $539$ as the result.

  Here you see the explanation:

  $49\times 11=49\times 10+49\times 1=490+49=539$

  $~$

  Squaring any number ending in $5$

  You still have to use FOIL multiplication for two binomials:

  $\begin{array}{rcl} 65^2&=&(60+5)^2\\ &=&60^2+2\times 60\times 5+25\\ &=&6^2\times 100+6\times 100+25\\ &=&6\times100(6+1)+25\\ &=&6\times 7\times 100+25\\ &=&4200+25=4225 \end{array}$