Multi step conversion word problems

Multi step conversion word problems DEFAULT

2.7: Solving Multi-step Conversion Problems

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Multiple Conversions

Sometimes you will have to perform more than one conversion to obtain the desired unit. For example, suppose you want to convert 54.7 km into millimeters. We will set up a series of conversion factors so that each conversion factor produces the next unit in the sequence. We first convert the given amount in km to the base unit, which is meters. We know that 1,000 m =1 km.

Then we convert meters to mm, remembering that \(1\; \rm{mm}\) = \( 10^{-3}\; \rm{m}\).

Concept Map

Convert kilometers to meters to millimeters: use conversion factors 1000 meters per 1 kilometer and 1 millimeter per 0.001 meter


\[ \begin{align*} 54.7 \; \cancel{\rm{km}} \times \dfrac{1,000 \; \cancel{\rm{m}}}{1\; \cancel{\rm{km}}} \times \dfrac{1\; \cancel{\rm{mm}}}{\cancel{10^{-3} \rm{m}}} & = 54,700,000 \; \rm{mm} \\ &= 5.47 \times 10^7\; \rm{mm} \end{align*}\]

In each step, the previous unit is canceled and the next unit in the sequence is produced, each successive unit canceling out until only the unit needed in the answer is left.

Example \(\PageIndex{1}\): Unit Conversion

Convert 58.2 ms to megaseconds in one multi-step calculation.


Steps for Problem Solving

Unit Conversion

Identify the "given" information and what the problem is asking you to "find."

Given: 58.2 ms

Find: Ms

List other known quantities

\(1 ms = 10^{-3} s \)

\(1 Ms = 10^6s \)

Prepare a concept map.

Convert milliseconds to seconds to microseconds: use conversion factors 0.001 second per millisecond and 1 microsecond per 1 million seconds


\[ \begin{align} 58.2 \; \cancel{\rm{ms}} \times \dfrac{10^{-3} \cancel{\rm{s}}}{1\; \cancel{\rm{ms}}} \times \dfrac{1\; \rm{Ms}}{1,000,000\; \cancel{ \rm{s}}} & =0.0000000582\; \rm{Ms} \nonumber\\ &= 5.82 \times 10^{-8}\; \rm{Ms}\nonumber \end{align}\nonumber \]

Neither conversion factor affects the number of significant figures in the final answer.

Example \(\PageIndex{2}\): Unit Conversion

How many seconds are in a day?


Steps for Problem Solving

Unit Conversion

Identify the "given" information and what the problem is asking you to "find."

Given: 1 day

Find: s

List other known quantities.

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

Prepare a concept map.

Convert day to hour to minute to second: use conversion factors 24 hours per day, 60 minutes per hour, and 60 seconds per minute


\[1 \: \text{d} \times \frac{24 \: \text{hr}}{1 \: \text{d}}\times \frac{60 \: \text{min}}{1 \: \text{hr}} \times \frac{60 \: \text{s}}{1 \: \text{min}} = 86,400 \: \text{s} \nonumber\]

Exercise \(\PageIndex{1}\)

Perform each conversion in one multi-step calculation.

  1. 43.007 ng to kg
  2. 1005 in to ft
  3. 12 mi to km
Answer a
\(4.3007 \times 10^{-11} kg \)
Answer b
\(83.75\, ft\)
Answer c
\(19\, km\)

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A pharmacist dispenses drugs that have been prescribed by a doctor. Although that may sound straightforward, pharmacists in the United States must hold a doctorate in pharmacy and be licensed by the state in which they work. Most pharmacy programs require four years of education in a specialty pharmacy school. Pharmacists must know a lot of chemistry and biology so they can understand the effects that drugs (which are chemicals, after all) have on the body. Pharmacists can advise physicians on the selection, dosage, interactions, and side effects of drugs. They can also advise patients on the proper use of their medications, including when and how to take specific drugs properly. Pharmacists can be found in drugstores, hospitals, and other medical facilities. Curiously, an outdated name for pharmacist is chemist, which was used when pharmacists formerly did a lot of drug preparation, or compounding. In modern times, pharmacists rarely compound their own drugs, but their knowledge of the sciences, including chemistry, helps them provide valuable services in support of everyone’s health.

An older African-American woman talks to a Hispanic male pharmacist as he explains her prescription.


In multi-step conversion problems, the previous unit is canceled for each step and the next unit in the sequence is produced, each successive unit canceling out until only the unit needed in the answer is left.


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Video transcript

Tomas dropped off two packages to be shipped one package weighed 1.38 kilograms and the other package weighed 720 grams so the first one given kilograms and the second one to give us in grams what was the combined weight of both packages in grams so what I want to do is I'm going to convert both of these to grams and then add them up to get the combined weight well we already know the second one is its 720 grams but what's the first one in terms of grams well let's see one 0.38 I could write kilograms like that well if one kilogram is 1,000 grams and that's what the prefix kilo means it means a thousand well to go from one to 1.38 I'm multiplying by one point three eight so I'm also if I want in terms of grams I'll take a thousand grams and multiply by one point three eight so this is going to be one thousand times one point three eight is one thousand three hundred and eighty grams which I could have just denoted with a G but now let's add those together this is the first package and the second package is seven hundred and twenty grams and so if I were to add this is going to be let's see zero ten let's see four plus seven is 11 and then one plus one is two we get 2100 grams in total let's do another one of these unit conversion examples Julia and her friends are making kites out of paper for each kite they need a piece of paper that is 0.65 meters wide how many centimeters how many centimeters of paper will they need to make four kites so they tell us how wide the paper is in meters for each kite but they want the answer in centimeters and we went for four kinds so let's think about a little bit each kite is 0.65 meters 0.65 meters so to go from 1 to 0.65 you multiply by 0.65 on the meters and so to go to 100 centimeters to the number of centimeters you need for the width of a kite you'd similarly x 0.65 well 65 hundredths times 100 is going to be 65 65 centimetres per kite so this is 0.
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1.6 Unit Conversion Word Problems

Chapter 1: Algebra Review

One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.

The trick is to decide what fractions to multiply. If an expression is multiplied by 1, its value does not change. The number 1 can be written as a fraction in many different ways, so long as the numerator and denominator are identical in value. Note that the numerator and denominator need not be identical in appearance, but rather only identical in value. Below are several fractions, each equal to 1, where the numerator and the denominator are identical in value. This is why, when doing dimensional analysis, it is very important to use units in the setup of the problem, so as to ensure that the conversion factor is set up correctly.

 If 1 pound = 16 ounces, how many pounds are in 435 ounces?

\begin{array}{rrll} 435\text{ oz}&=&435\text{ \cancel{oz}}\times \dfrac{1\text{ lb}}{16\text{ \cancel{oz}}} \hspace{0.2in}& \text{This operation cancels the oz and leaves the lbs} \\ \\ &=&\dfrac{435\text{ lb}}{16} \hspace{0.2in}& \text{Which reduces to } \\ \\ &=&27\dfrac{3}{16}\text{ lb} \hspace{0.2in}& \text{Solution} \end{array}

The same process can be used to convert problems with several units in them. Consider the following example.

A student averaged 45 miles per hour on a trip. What was the student’s speed in feet per second?

\begin{array}{rrll} 45 \text{ mi/h}&=&\dfrac{\text{45 \cancel{mi}}}{\text{\cancel{hr}}}\times \dfrac{5280 \text{ ft}}{1\text{ \cancel{mi}}}\times \dfrac{1\text{ \cancel{hr}}}{3600\text{ s}}\hspace{0.2in}&\text{This will cancel the miles and hours} \\ \\ &=&45\times \dfrac{5280}{1}\times \dfrac{1}{3600} \text{ ft/s}\hspace{0.2in}&\text{This reduces to} \\ \\ &=&66\text{ ft/s}\hspace{0.2in}&\text{Solution} \end{array}

Convert 8 ft3 to yd3.

\begin{array}{rrll} 8\text{ ft}^3&=&8\text{ ft}^3 \times \dfrac{(1\text{ yd})^3}{(3\text{ ft})^3}&\text{Cube the parentheses} \\ \\ &=&8\text{ }\cancel{\text{ft}^3}\times \dfrac{1\text{ yd}^3}{27\text{ }\cancel{\text{ft}^3}}&\text{This will cancel the ft}^3\text{ and replace them with yd}^3 \\ \\ &=&8\times \dfrac{1\text{ yd}^3}{27}&\text{Which reduces to} \\ \\ &=&\dfrac{8}{27}\text{ yd}^3\text{ or }0.296\text{ yd}^3&\text{Solution} \end{array}

A room is 10 ft by 12 ft. How many square yards are in the room? The area of the room is 120 ft2 (area = length × width).

Converting the area yields:

\begin{array}{rrll} 120\text{ ft}^2&=&120\text{ }\cancel{\text{ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ }\cancel{\text{ft}})^2}&\text{Cancel ft}^2\text{ and replace with yd}^2 \\ \\ &=&\dfrac{120\text{ yd}^2}{9}&\text{This reduces to} \\ \\ &=&13\dfrac{1}{3}\text{ yd}^2&\text{Solution} \\ \\ \end{array}

The process of dimensional analysis can be used to convert other types of units as well. Once relationships that represent the same value have been identified, a conversion factor can be determined.

A child is prescribed a dosage of 12 mg of a certain drug per day and is allowed to refill his prescription twice. If there are 60 tablets in a prescription, and each tablet has 4 mg, how many doses are in the 3 prescriptions (original + 2 refills)?

\begin{array}{rrll} 3\text{ prescriptions}&=&3\text{ \cancel{pres.}}\times \dfrac{60\text{ \cancel{tablets}}}{1\text{ \cancel{pres.}}}\times \dfrac{4\text{ \cancel{mg}}}{1\text{ \cancel{tablet}}}\times \dfrac{1\text{ dosage}}{12\text{ \cancel{mg}}}&\text{This cancels all unwanted units} \\ \\ &=&\dfrac{3\times 60\times 4\times 1}{1\times 1\times 12}\text{ or }\dfrac{720}{12}\text{ dosages}&\text{Which reduces to} \\ \\ &=&60\text{ daily dosages}&\text{Solution} \\ \\ \end{array}


    \[\begin{array}{rrlrrl} 12\text{ in}&=&1\text{ ft}\hspace{1in}&10\text{ mm}&=&1\text{ cm} \\ 3\text{ ft}&=&1\text{ yd}&100\text{ cm}&=&1\text{ m} \\ 1760\text{ yds}&=&1\text{ mi}&1000\text{ m}&=&1\text{ km} \\ 5280\text{ ft}&=&1\text{ mi}&&& \end{array}\]

Imperial to metric conversions:

    \[\begin{array}{rrl} 1\text{ inch}&=&2.54\text{ cm} \\ 1\text{ ft}&=&0.3048\text{ m} \\ 1\text{ mile}&=&1.61\text{ km} \end{array}\]


    \[\begin{array}{rrlrrl} 144\text{ in}^2&=&1\text{ ft}^2\hspace{1in}&10,000\text{ cm}^2&=&1\text{ m}^2 \\ 43,560\text{ ft}^2&=&1\text{ acre}&10,000\text{ m}^2&=&1\text{ hectare} \\ 640\text{ acres}&=&1\text{ mi}^2&100\text{ hectares}&=&1\text{ km}^2 \end{array}\]

Imperial to metric conversions:

    \[\begin{array}{rrl} 1\text{ in}^2&=&6.45\text{ cm}^2 \\ 1\text{ ft}^2&=&0.092903\text{ m}^2 \\ 1\text{ mi}^2&=&2.59\text{ km}^2 \end{array}\]


    \[\begin{array}{rrlrrl} 57.75\text{ in}^3&=&1\text{ qt}\hspace{1in}&1\text{ cm}^3&=&1\text{ ml} \\ 4\text{ qt}&=&1\text{ gal}&1000\text{ ml}&=&1\text{ litre} \\ 42\text{ gal (petroleum)}&=&1\text{ barrel}&1000\text{ litres}&=&1\text{ m}^3 \end{array}\]

Imperial to metric conversions:

    \[\begin{array}{rrl} 16.39\text{ cm}^3&=&1\text{ in}^3 \\ 1\text{ ft}^3&=&0.0283168\text{ m}^3 \\ 3.79\text{ litres}&=&1\text{ gal} \end{array}\]


    \[\begin{array}{rrlrrl} 437.5\text{ grains}&=&1\text{ oz}\hspace{1in}&1000\text{ mg}&=&1\text{ g} \\ 16\text{ oz}&=&1\text{ lb}&1000\text{ g}&=&1\text{ kg} \\ 2000\text{ lb}&=&1\text{ short ton}&1000\text{ kg}&=&1\text{ metric ton} \end{array}\]

Imperial to metric conversions:

    \[\begin{array}{rrl} 453\text{ g}&=&1\text{ lb} \\ 2.2\text{ lb}&=&1\text{ kg} \end{array}\]


Fahrenheit to Celsius conversions:

    \[\begin{array}{rrl} ^{\circ}\text{C} &= &\dfrac{5}{9} (^{\circ}\text{F} - 32) \\ \\ ^{\circ}\text{F}& =& \dfrac{9}{5}(^{\circ}\text{C} + 32) \end{array}\]

Fahrenheit to Celsius conversion scale. Long description available.

For questions 1 to 18, use dimensional analysis to perform the indicated conversions.

  1. 7 miles to yards
  2. 234 oz to tons
  3. 11.2 mg to grams
  4. 1.35 km to centimetres
  5. 9,800,000 mm to miles
  6. 4.5 ft2 to square yards
  7. 435,000 m2 to square kilometres
  8. 8 km2 to square feet
  9. 0.0065 km3 to cubic metres
  10. 14.62 in3 to square centimetres
  11. 5500 cm3 to cubic yards
  12. 3.5 mph (miles per hour) to feet per second
  13. 185 yd per min. to miles per hour
  14. 153 ft/s (feet per second) to miles per hour
  15. 248 mph to metres per second
  16. 186,000 mph to kilometres per year
  17. 7.50 tons/yd2 to pounds per square inch
  18. 16 ft/s2 to kilometres per hour squared

For questions 19 to 27, solve each conversion word problem.

  1. On a recent trip, Jan travelled 260 miles using 8 gallons of gas. What was the car’s miles per gallon for this trip? Kilometres per litre?
  2. A certain laser printer can print 12 pages per minute. Determine this printer’s output in pages per day.
  3. An average human heart beats 60 times per minute. If the average person lives to the age of 86, how many times does the average heart beat in a lifetime?
  4. Blood sugar levels are measured in milligrams of glucose per decilitre of blood volume. If a person’s blood sugar level measured 128 mg/dL, what is this in grams per litre?
  5. You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?
  6. A cargo container is 50 ft long, 10 ft wide, and 8 ft tall. Find its volume in cubic yards and cubic metres.
  7. A local zoning ordinance says that a house’s “footprint” (area of its ground floor) cannot occupy more than ¼ of the lot it is built on. Suppose you own a \frac{1}{3}-acre lot (1 acre = 43,560 ft2). What is the maximum allowed footprint for your house in square feet? In square metres?
  8. A car travels 23 km in 15 minutes. How fast is it going in kilometres per hour? In metres per second?
  9. The largest single rough diamond ever found, the Cullinan Diamond, weighed 3106 carats. One carat is equivalent to the mass of 0.20 grams. What is the mass of this diamond in milligrams? Weight in pounds?

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-1-6/”>Answer Key 1.6

Celsius to Fahrenheit conversion scale long description: Scale showing conversions between Celsius and Fahrenheit. The following table summarizes the data:


[Return to Celsius to Fahrenheit conversion scale]

Multi-step unit conversion word problem - Introduction to algebra - Algebra I - Khan Academy

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Unit Conversions: Multi-Step Problems (Honors Chemistry Only)

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