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1.2: Problem Solving and Estimating

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    1.2 Learning Objectives

    • Apply the problems solving process to real life problems
    • Use dimensional analysis to solve problems
    • Use estimates to solve real life problems
    • Round decimals

    In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes.

    This approach does not work well with real life problems. Instead, problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway, a series of steps that will allow you to answer the question.

    Problem Solving Process

    1. Identify the question you’re trying to answer.
    2. Work backwards, identifying the information you will need and the relationships you will use to answer that question.
    3. Continue working backwards, creating a solution pathway.
    4. If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
    5. Solve the problem, following your solution pathway.
    6. Check your answer. Does the answer make sense?

    In most problems we work, we will be estimating a solution, because we will not have perfect information. We will begin with a few examples where we will be able to estimate the solution using basic knowledge from our lives.

    Definition: Estimation

    Estimation is the process of arriving at an approximate answer to a question. Usually by rounding answers to a place value that is easier to express in the situation.

    Depending on the situation, you may be asked to round your answers to a specific place value. The tables below list more common values, but there are many more. 

    Place values of a whole number:

    An image of a table of place Values of Whole Numbers, namely ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, billions, ten billions, hundred billions, trillions, ten trillions, and hundred trillions

    Place values of a decimal number:

                                A table of decimal place values beginning with tens and ones on the left of the decimal, then the decimal point, then tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths

    Name and Write Decimals

    Notice that “ten thousand” is a number larger than one, but “one ten-thousandth” is a number smaller than one. The “th” at the end of the name tells you that the number is smaller than one.

    When we name a whole number, the name corresponds to the place value based on the powers of ten. We read 10,000 as “ten thousand” and 10,000,000 as “ten million.” Likewise, the names of the decimal places correspond to their fraction values.

    We summarize the steps needed to name a decimal below.

    1. Name the number to the left of the decimal point.
    2. Write “and” for the decimal point.
    3. Name the “number” part to the right of the decimal point as if it were a whole number.
    4. Name the decimal place of the last digit.

    Example 1

    Name the decimal \(4.3\).

    Solution

    A table is given with four steps. Additionally, the number 4.3 is given. The first step reads “Step 1. Name the number to the left of the decimal point.” To the right of this, it is noted that “4 is to the left of the decimal point.” To the right of this, it reads “four” followed by a large blank space.The second step reads “Step 2. Write ‘and’ for the decimal point.” To the right of this it reads “four and” followed by a blank space.The third step reads “Step 3. Name the ‘number’ part to the right of the decimal point as if it were a whole number.” To the right of this, it reads “3 is to the right of the decimal point.” To the right of this, it reads “four and three” followed by a blank.Finally, the last step reads “Step 4. Name the decimal place.” To the right of this, it reads “four and three tenths.”

    Try It Now 1

    Name the decimal \(6.7\).

    Answer

    six and seven tenths

    Try It Now 2

    Name the decimal: \(−15.571\).

    Answer
      \(−15.571\)
    Name the number to the left of the decimal point. negative fifteen __________________________________
    Write “and” for the decimal point. negative fifteen and ______________________________
    Name the number to the right of the decimal point. negative fifteen and five hundred seventy-one __________
    The \(1\) is in the thousandths place. negative fifteen and five hundred seventy-one thousandths

    When we write or read a check we interpret both the numerals and the name of the number. Let’s see how to write the decimal from the name.

    We summarize the steps to writing a decimal.

    1. Look for the word “and”—it locates the decimal point.
      • Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point.
      • If there is no “and,” write a “0” with a decimal point to its right.
    2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
    3. Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
    4. Fill in zeros for place holders as needed.

    Example 2

    Write “fourteen and twenty-four thousandths” as a decimal.

    Solution

    A table is given with four steps. The first step reads “Step 1. Look for the work ‘and’ – it locates the decimal point. Place a decimal point under the word ‘and’. Translate the words before ‘and’ into the whole number and place it to the left of the decimal point.” To the right of this, we have the words “fourteen and twenty-four thousandths.” Below this word, we have “fourteen and twenty-four thousandths” with the word “and” underlined. Below this word, we have a small blank space separated from a larger blank space by a decimal point. Under this, we have 14 in the small blank space followed by the decimal point and the larger blank space.The second step reads “Step 2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.” To the right of this it reads “The last word is thousandths.” To the right of this there is the number 14 followed by a decimal point and three small blank spaces. Under the blank spaces, the words “tenths,” “hundredths,” and “thousandths” are written.The third step reads “Step 3. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces – putting the final digit in the last place.” To the right of this, we have 14 followed by a decimal followed by a blank space followed by 2 and 4 on the other two previously blank spaces.Finally, the last step reads “Step 4. Fill in zeros for empty place holders as needed.” To the right of this, it reads “Zeros are needed in the tenths place.” To the right of this, we have 14 followed by a decimal point followed by 0, 2, and 4, respectively, on the blank spaces. Below this, we have “fourteen and twenty-four thousandths is written 14.024.”

    Try It Now 3

    Write as a decimal: thirteen and sixty-eight thousandths.

    Answer

    13.068

    Round Decimals

    Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.

    We summarize the steps for rounding a decimal here.

    1. Locate the given place value and mark it with an arrow.
    2. Underline the digit to the right of the place value.
    3. Is this digit greater than or equal to 5?
      • Yes—add 1 to the digit in the given place value.
      • No—do not change the digit in the given place value.
    4. Rewrite the number, deleting all digits to the right of the rounding digit.

    Example 3

    Round 18.379 to the nearest hundredth.

    Solution

    A table is given with four steps. The first step reads “Step 1: Locate the given place value and mark it with an arrow.” To the right of this, we have the number 18.379; above it, are the words hundreds place, which has an arrow pointing to the 7.The second step reads “Step 2. Underline the digit to the right of the given place value.” To the right of this, we have 18.379 with the 9 underlined.The third step reads “Step 3. Is this digit greater than or equal to 5? Below this reads, “Yes: add 1 to the digit in the given place value.” Below this reads, “No: do not change the digit in the given place value.” To the right of this, it says “Because 9 is greater than or equal to ” To the right of this, we have the number 18.379 with the 9 marked “delete” and the 7 marked “add 1.”Finally, the last step reads “Step 4. Rewrite the number, removing all digits to the right of the rounding digit.” To the right of this, we have 18.38 followed by “18.38 is 18.379 rounded to the nearest hundredth.”

    Try It Now 4

    Round 6.582 to the nearest

    1. hundredth
    2. tenth
    3. whole number.
    Answer
    1. 6.58
    2. 6.6
    3. 7

    We can use these estimation techniques to answer questions that involve some problem solving. Bringing the two concepts from above together, we apply them to several examples below.

    Example 4

    How many times does your heart beat in a year?

    Solution

    This question is asking for the rate of heart beats per year. Since a year is a long time to measure heart beats for, if we knew the rate of heart beats per minute, we could scale that quantity up to a year. So the information we need to answer this question is heart beats per minute. This is something you can easily measure by counting your pulse while watching a clock for a minute.

    Suppose you count 80 beats in a minute. To convert this beats per year:

    \(\dfrac{80 \text { beats }}{1 \cancel {\text { minute }}} \cdot \dfrac{60 \cancel {\text { minutes }}}{1 \cancel {\text { hour }}} \cdot \dfrac{24 \cancel {\text { hours }}}{1 \cancel {\text { day }}} \cdot \dfrac{365 \cancel{ \text { days }}}{1 \text { year }}=42,048,000\) beats per year

    Example 5

    How thick is a single sheet of paper? How much does it weigh?

    Solution

    While you might have a sheet of paper handy, trying to measure it would be tricky. Instead we might imagine a stack of paper, and then scale the thickness and weight to a single sheet. If you’ve ever bought paper for a printer or copier, you probably bought a ream, which contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick and weighs about 5 pounds. Scaling these down,

    \(\dfrac{2 \text { inches }}{ \cancel{\text { ream }}} \cdot \dfrac{1 \cancel{ \text { ream }}}{500 \text { pages }}=0.004\) inches per sheet

    \(\dfrac{5 \text { pounds }}{ \cancel {\text { ream }}} \cdot \dfrac{1 \cancel {\text { ream }}}{500 \text { pages }}=0.01\) pounds per sheet, or 0.16 ounces per sheet.

    Example 6

    A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

    Solution

    There are several possible solution pathways to answer this question. We will explore one.

    To answer the question of how many calories 4 mini-muffins will contain, we would want to know the number of calories in each mini-muffin. To find the calories in each mini-muffin, we could first find the total calories for the entire recipe, then divide it by the number of mini-muffins produced. To find the total calories for the recipe, we could multiply the calories per standard muffin by the number per muffin. Notice that this produces a multi-step solution pathway. It is often easier to solve a problem in small steps, rather than trying to find a way to jump directly from the given information to the solution.

    We can now execute our plan:

    \(12 \cancel{\text { muffins}} \cdot \dfrac{250 \text { calories }}{ \cancel {\text { muffin }}}=3000\) calories for the whole recipe

    \(\dfrac{3000 \text { calories }}{20 \text { mini-muffins }}\) gives 150 calories per mini-muffin

    \(4 \cancel {\text { mini-muffins }} \cdot \dfrac{150 \text { calories }}{ \cancel {\text { mini-muffin }}}\) totals 600 calories consumed.

    Example 7

    You need to replace the boards on your deck. About how much will the materials cost?

    Solution

    There are two approaches we could take to this problem: 1) estimate the number of boards we will need and find the cost per board, or 2) estimate the area of the deck and find the approximate cost per square foot for deck boards. We will take the latter approach.

    For this solution pathway, we will be able to answer the question if we know the cost per square foot for decking boards and the square footage of the deck. To find the cost per square foot for decking boards, we could compute the area of a single board, and divide it into the cost for that board. We can compute the square footage of the deck using geometric formulas. So first we need information: the dimensions of the deck, and the cost and dimensions of a single deck board.

    Suppose that measuring the deck, it is rectangular, measuring 16 ft by 24 ft, for a total area of \(384 \mathrm{ft}^{2}\).

    From a visit to the local home store, you find that an 8 foot by 4 inch cedar deck board costs about $7.50. The area of this board, doing the necessary conversion from inches to feet, is:

    \(8 \text { feet } \cdot 4 \cancel { \text{inches } }\cdot \dfrac{1 \text { foot }}{12 \cancel {\text{ inches} }}=2.667 \mathrm{ft}^{2}\).

    The cost per square foot is then

    \(\dfrac{\$ 7.50}{2.667 \mathrm{ft}^{2}}=\$ 2.8125 \text { per } \mathrm{ ft}^{2}\).

    This will allow us to estimate the material cost for the whole \(384 \mathrm{ ft}^{2}\) deck

    \(\ 384 \cancel {\mathrm{ ft}^{2} }\cdot \dfrac{\$ 2.8125}{\cancel{\mathrm{ ft}^{2}}}=\$ 1080\) total cost.

    Of course, this cost estimate assumes that there is no waste, which is rarely the case. It is common to add at least 10% to the cost estimate to account for waste.

    Example 8

    Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

    Solution

    To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.

    It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.

    From Hyundai’s website, the 2013 Sonata will get 24 miles per gallon (mpg) in the city, and 35 mpg on the highway. The hybrid will get 35 mpg in the city, and 40 mpg on the highway.

    An average driver drives about 12,000 miles a year. Suppose that you expect to drive about 75% of that in the city, so 9,000 city miles a year, and 3,000 highway miles a year.

    We can then find the number of gallons each car would require for the year.

    Sonata:

    \(9000 \cancel {\text { city miles } }\cdot \dfrac{1 \text { gallon }}{24 \cancel{ \text { city miles }}}+3000 \cancel{ \text { hightway miles}}. \dfrac{1 \text { gallon }}{35 \cancel{ \text { highway miles }}}=460.7\text{ gallons}\)

    Hybrid:

    \(9000 \cancel {\text { city miles }}\cdot \dfrac{1 \text { gallon }}{35 \cancel{ \text { city miles }}}+3000 \cancel {\text { highway miles}}. \dfrac{1 \text { gallon }}{40 \cancel {\text { highway miles }}}=332.1\text{ gallons}\)

    If gas in your area averages about $3.50 per gallon, we can use that to find the running cost:

    Sonata: \(460.7 \text { gallons } \cdot \dfrac{\$ 3.50}{\text { gallon }}=\$ 1612.45\)

    Hybrid: \(\text { 332.1 gallons } \cdot \dfrac{\$ 3.50}{\text { gallon }}=\$ 1162.35\)

    The hybrid will save $450.10 a year. The gas costs for the hybrid are about \(\dfrac{\$ 450.10}{\$ 1612.45} = 0.279 = 27.9\%\) lower than the costs for the standard Sonata.

    While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about $25,850, compared to the base model for the regular Sonata, at $20,895.

    To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs $4965 more. With gas savings of $451.10 a year, it will take about 11 years for the gas savings to make up for the higher initial costs.

    We can conclude that if you expect to own the car 11 years, the hybrid is indeed worth it. If you plan to own the car for less than 11 years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.

    Try it Now 5

    If traveling from Seattle, WA to Spokane WA for a three-day conference, does it make more sense to drive or fly?

    Answer

    There is not enough information provided to answer the question, so we will have to make some assumptions, and look up some values.

    Assumptions:

    a) We own a car. Suppose it gets 24 miles to the gallon. We will only consider gas cost.

    b) We will not need to rent a car in Spokane, but will need to get a taxi from the airport to the conference hotel downtown and back.

    c) We can get someone to drop us off at the airport, so we don’t need to consider airport parking.

    d) We will not consider whether we will lose money by having to take time off work to drive.

    Values looked up (your values may be different)

    a) Flight cost: \(\$184\)

    b) Taxi cost: \(\$25\) each way (estimate, according to hotel website)

    c) Driving distance: \(280\) miles each way

    d) Gas cost: \(\$3.79\) a gallon

    Cost for flying: \(\$184\text{ flight cost }+ \$50\text{ in taxi fares }= \$234\).

    Cost for driving: \(560\) miles round trip will require 23.3 gallons of gas, costing \(\$88.31\).

    Based on these assumptions, driving is cheaper. However, our assumption that we only include gas cost may not be a good one. Tax law allows you deduct \(\$0.55\) (in 2012) for each mile driven, a value that accounts for gas as well as a portion of the car cost, insurance, maintenance, etc. Based on this number, the cost of driving would be \(\$319\).

    This page titled 1.2: Problem Solving and Estimating is shared under a CC BY-SA license and was authored, remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.