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Problem Solving Plan, Mental Math

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Problem Solving Plan, Mental Math

Credit: Kecko
Source: https://www.flickr.com/photos/kecko/6813792712/in/photolist-bo7tbG-fuLzA-5y14tD-dvedPY-6amFKn-5KrZse-9gHjBc-kJ1Z8M-mCSV8a-qu92DD-9GmEK5-a1kxy8-yuyTG4-75jzKX-fJPfHp-bJweHt-fuKSD-6xs5mp-enY3qD-5CVq7X-dMJLtb-781nU-qGVoP9-5kuKZ4-cBHZPL-fuLzx-5VzQ5Y-6nBe4o-5JZF2z-5mLuHu-5LdT3B-5KrZsK-LonYs-omUcxZ-fGEU7S-2cR3Q8-gf3PQB-nELdFT-gf4pZM-oD7Vug-83WhRW-7dqviq-dv8Hp8-oKU6NR-a132K8-a4eYDF-byhkqp-3XdpMd-4VxhQf-9aBbP
License: CC BY-NC 3.0

Sayber and Tuscany ride the city bus to school every weekday. The bus charges $2 per passenger, and, with the student discount they get, the kids pay another $3 each for the extended ride.

If the kids’ parents give them enough money at the beginning of the week to cover bus fare, how much do they receive together each week?

In this concept, you will learn about using mental math to solve word problems.

Problem Solving

There are a number of common ways to solve word problems, and practicing each of them is a great idea, since a plan that works perfectly in one situation my not be so great for another. The Guess and Check and the Work Backward models are great problem solving plans, but sometimes it makes more sense to use mental math.

By applying mental math to a problem, you can often solve it as quickly (and more efficiently) as you could have by guessing. The trick to using mental math is to look for more and easier operations rather than fewer but more complex ones.

For example, consider this multiplication problem:

Kayte has 14 boxes of popcorn, each with 7 packages inside. How many packages does she have in total?

You could solve this using vertical multiplication on paper, or even by guessing and then dividing to check, but even though you may not know what 14 times 7 is right away, you can break in into two more easily computed multiplication problems.

\begin{align*}14 \text { is the same as } 10 + 4 \\ 10 \text { times } 7 = 70 \\ 4 \text { times } 7 = 28 \\ 70 + 28 = 98 \\ 70 + 20 = 90 \\ 90 + 8 = 98\end{align*} 

Even though you end up with more steps by splitting up the 14 before you multiply, the math is easier this way and you are more likely to be able to quickly compute it in your head.


Example 1

Earlier, you were given a problem about Sayber and Tuscany each receiving $2 to board the bus and $3 for the ride each day for 5 days.

You can use mental math to calculate the total amount of money their parents give them.

First, consider the problem.

2 kids times ($2 + $3) times 5 days

Next, break the problem into smaller steps that are easier to calculate in your head. 2 + 3 is easy, that’s 5. Now you have \begin{align*}2 \times 5 \times 5\end{align*}.

\begin{align*}2 \times 5 = 10\end{align*}, so that would leave \begin{align*}10 \times 5\end{align*}.

Remember, you can multiply by 10 by just adding a zero, so \begin{align*}10 \times 5 = 50\end{align*}.

Sayber and Tuscany should receive a total of $50 from their parents for bus fares each school week.

Example 2

Ground beef for hamburgers is sold by the pound. If Delilah buys 12 pounds at $2.50 per pound, how much did she spend?

First, consider the problem and see if it can be split into easily-computed pieces.

The hamburger is $2.50 per pound, and Delilah is buying 12 pounds, so you need to multiply 12 by 2.5. You may not know what 12 times 2.5 is right away, but 12 is the same as 10 + 2, and multiplying by 10 is simple as you just add a zero. Then, you can just add 2.5 times 2 to the answer.

Next, compute each piece separately using mental math.

\begin{align*}2.5 \times 10 &= 25 \text { (Multiply by ten easily by moving the decimal one place to the right.) }\\ 2.5 \times 2 &= \hphantom{0}5 \\ 25 + 5 &=30\end{align*} 

The answer is Delilah will spend a total of $30 on 12 pounds of hamburger, at $2.50 per pound.

Example 3

Ravi lives in Florida and loves going to pick strawberries during strawberry season. He can fit 20 strawberries in one box. If he is given a barrel of 500 strawberries, how many boxes will it take for Ravi to sort the strawberries?

First, consider the problem to see if there is a way to break it into smaller pieces that you may be able to calculate quickly in your head.

In this case, you need to divide 500 by 20. You may not know right away what 500 divided by 20 is, but dividing by 10 is always easy, you just remove a zero. Then you can just divide that answer in half, since there can only be half as many 20-strawberry boxes as 10-strawberry boxes.

Next, handle each mental calculation separately.

\begin{align*}500 \text { divided by } 10 = 50\\ 50 \text { divided by } 2 = 25\end{align*}

The answer is Ravi will need 25 boxes of 20 strawberries each in order to pack 500 strawberries total.

Example 4

If Ravi fills 40 boxes during his shift at work, how many strawberries did he start with?

First, consider the problem and see if it can be broken into smaller bits.

Ravi puts 20 strawberries into each box, so if he fills 40 boxes, that is 20 times 40 strawberries total. You may not know what 20 times 40 is right away, but 10 times 40 is easy, and 20 times 40 is just twice that many.

Next, use mental math to multiply each of the smaller parts of the overall problem.

\begin{align*}10 \text { times } 40 &= 400\\ 400 \text { times } 2 &= 800\\ \text { If } 4 \text { times } 2 = 8, \text { then } 400 \text { times } 2 &= 800\end{align*}  

The answer is if Ravi fills 40 boxes of 20 strawberries each, he packed 800 strawberries.

Example 5

If Ravi works two shifts, at this rate, how many boxes will he fill?

This one is easy, since it is only one step to start with!

40 boxes per shift times 2 shifts is 80 boxes total

The answer is 80 boxes.


Use mental math to solve each of the following problems.

  1. Josie helps label pints of strawberries. She labels them at \begin{align*}\$2.00\end{align*} per pint. If she sells 100 pints, how much money has she made?
  2. If Josie sells $400.00 worth of strawberries, how many pints has she sold?
  3. Josie also enjoys making strawberry milkshakes. If it takes 5 strawberries to make one milkshake, how many can she make with 20 strawberries?
  4. If Josie makes 35 strawberry milkshakes in one day, how many strawberries does she need to accomplish this task?
  5. If there are 25 strawberries in a pint, how many pints does Josie use to make her 35 milkshakes?
  6. Carl loves to collect old vinyl records. He has a whole collection that he received from his Dad. If he has five different categories of records with twenty records in each category, how many records does Carl have altogether?
  7. Julie is a friend of Carl’s. She brought over her collection of records. Julie has 254 records. If she and Carl were to combine their collections, how many would they have altogether?
  8. When Carl and his Mom went to a yard sale, Carl got a box of vinyl records for $25.00. He brought them home and looked in the box. Out of 30 records, five of them were broken or scratched. If he puts these new records with his collection, how many does he now have?
  9. Carl’s sister borrowed ten records to show her friend. How many are left in Carl’s collection?
  10. Mario is an outstanding skateboarder. He recently purchased a new skateboard. He wants to sell his old one. A friend wishes to buy it for \begin{align*}\$45\end{align*}. If he gives Mario three twenty dollar bills, how much change should Mario give his friend?
  11. If Mario buys a new skateboard for double the price that he sold his old one, how much did he pay for the new skateboard?
  12. If Mario has \begin{align*}\$100.00\end{align*} and he buys the skateboard for double the price that he sold his old one, does he have enough money to make the purchase?
  13. Did he receive any change back at the skateboard shop? How much?
  14. Karen collects fairy figurines. She was given 3 for her birthday, 2 for Christmas, 4 from her grandmother and 3 she bought on her own. How many fairy figurines does she have in all?
  15. Karen’s little sister loves one of the figurines. Karen has decided to give her the little fairy as a gift. After she does this, how many figurines will Karen have left?
  16. At a yard sale, Karen bought 5 fairies for \begin{align*}\$20.00\end{align*}. How much did she pay per fairy?
  17. How many fairies does Karen have now?
  18. Karen’s friend Emily also collects fairies. If Emily has twice as many fairies as Karen, how many does she have?
  19. Jamie runs track at school. He is one of the fastest runners on the team and runs one mile in about 5 minutes. How long will it take Jamie to run 10 miles?
  20. If Jamie runs a 3 mile race, about how much time will it take to run the 3 miles at his one mile pace?

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.18.


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Difference The result of a subtraction operation is called a difference.
Product The product is the result after two amounts have been multiplied.
Quotient The quotient is the result after two amounts have been divided.
Sum The sum is the result after two or more amounts have been added together.
Word Problem A word problem is a problem that uses verbal language to explain a mathematical situation.

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