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Conversion Factors: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 The matching measures

For a model kit, one rail is exactly four links long. A proposed conversion factor has $12$ links in its numerator. What number of rails belongs in its denominator so that the factor equals one?
1. Hint 1
  The numerator and denominator must describe equal lengths.
2. Hint 2
  Group the twelve links into groups matching one rail each.
Answer

$3$ rails.

Full solution

Each rail contains four links, so the length of twelve links in rails is
$12\div4=3$
Thus twelve links and three rails are equal lengths.

The factor is twelve links divided by three rails.
Its top and bottom name the same length, so the factor equals one.

Answer

$3$ rails.

Key idea

A conversion factor equals one when its numerator and denominator represent the same amount.
Problem 1: Mark complete
Problem 2 The label multiplier

A label shows how many $5$ -centimeter pieces make up a length of $x$ meters. Write that number as a single multiple of $x$ . Use $1$ meter equal to $100$ centimeters.
1. Hint 1
  One piece and five centimeters are equal lengths, so they can form a conversion factor too.
2. Hint 2
  Build a factor with centimeters on top, then one with pieces on top and centimeters below.
Answer

$20x$ pieces.

Full solution

Meters must cancel, so the first factor has centimeters on top and meters below, and $x$ meters becomes $100x$ centimeters.

Centimeters must cancel next, so the second factor has one piece on top and five centimeters below.
It divides the number of centimeters by five:
$\frac{100x}{5}=20x$
Together the two factors multiply $x$ by $100$ times one fifth, which is $20$ .

For example, one meter is one hundred centimeters, which makes twenty pieces, matching $20x$ when $x=1$ .

Answer

$20x$ pieces.

Key idea

Converting and then regrouping is a chain of two factors, and the overall multiplier is their product.
Problem 2: Mark complete
Problem 3 The wall panels

Two wall panels each cover $2.5$ square feet. The gap between them is a strip of $108$ square inches. Find the total area of the two panels and the strip, in square feet. Use $1$ foot equal to $12$ inches.
1. Hint 1
  Areas can be added only once they are measured in the same square unit.
2. Hint 2
  A square inch hides two lengths, so the inch-to-foot factor is used twice.
3. Hint 3
  Square inches must cancel, so the length factor carries inches on the bottom.
Answer

$5.75$ square feet, or $5\tfrac34$ square feet.

Full solution

The strip must change from square inches to square feet.
Inches must cancel, so the length factor is one foot per twelve inches, and a square unit needs it twice: one square foot is $12\cdot12=144$ square inches.

The strip is therefore
$108\div144=0.75$
square feet.
Dividing by $12$ only once would give $9$ square feet, the error of converting a square unit once.

The total area is
$2(2.5)+0.75=5.75$
square feet.

As a check in square inches, each panel is $2.5\cdot144=360$ square inches, so the total is $2(360)+108=828$ square inches, and $828\div144=5.75$ square feet.

Answer

$5.75$ square feet, or $5\tfrac34$ square feet.

Key idea

Areas can be added once they share one square unit, and converting to a larger square unit divides the number by the square of the length ratio, here $12$ squared, or $144$ .
Problem 3: Mark complete
Problem 4 The supply cartons

A craft center receives $3$ cartons of ribbon. Each carton holds $4$ rolls, each roll is $2$ yards long, one yard is $3$ feet, and one foot is $12$ inches. It sets aside $48$ inches. How many inches remain? Explain the unit cancellations in your conversion chain.
1. Hint 1
  Follow the quantities from cartons to rolls to yards to feet to inches.
2. Hint 2
  Each conversion factor puts the current unit below and the next unit above.
Answer

$816$ inches.

Full solution

Use factors of four rolls per carton, two yards per roll, three feet per yard, and twelve inches per foot.
Cartons, rolls, yards, and feet cancel successively, leaving inches.

The numerical product is
$3\cdot4\cdot2\cdot3\cdot12=864$
After setting aside the ribbon,
$864-48=816$
So $816$ inches remain.

As a check, grouping the same product differently, the $3$ cartons hold $12$ rolls, each roll is $2\cdot3\cdot12=72$ inches, and $12\cdot72=864$ inches, the same delivery total.

Answer

$816$ inches.

Key idea

A chain of true factors carries a quantity through its intermediate units, and amounts can be combined once they share a unit.
Problem 4: Mark complete
Problem 5 The pair of dispensers

Dispenser A releases $2.7$ liters per hour. Dispenser B releases $0.8$ milliliters per second. Use $1$ liter equal to $1000$ milliliters, $1$ hour equal to $60$ minutes, and $1$ minute equal to $60$ seconds. Which dispenser releases more, and by how many milliliters per minute?
1. Hint 1
  Two rates can be compared directly only when they use the same volume unit and the same time unit.
2. Hint 2
  Convert each rate to milliliters per minute. For A, change liters to milliliters as well as hours to minutes.
3. Hint 3
  A time unit in the denominator of a rate cancels against a factor that carries that unit on top.
Answer

Dispenser B, by $3$ milliliters per minute.

Full solution

For A, the factor of one thousand milliliters per liter cancels liters on top.
Hours sit in the denominator, so the time factor is one hour per sixty minutes, with hours on top.
The rate is
$\frac{2.7(1000)}{60}=45$
milliliters per minute.

For B, seconds sit in the denominator, so the time factor is sixty seconds per minute, with seconds on top.
The rate is
$0.8(60)=48$
milliliters per minute.

So Dispenser B releases more, by $48-45=3$ milliliters per minute.

As a check in milliliters per second, A releases $45\div60=0.75$ , which is less than B's $0.8$ .

Answer

Dispenser B, by $3$ milliliters per minute.

Key idea

Rates become directly comparable after both numerator and denominator units agree.
Problem 5: Mark complete
Problem 6 The water bottles

A tank holds $0.048$ cubic meters of water. All of it is poured into bottles that each hold $1.5$ liters. How many bottles does the water fill exactly? Use $1$ meter equal to $100$ centimeters and $1$ liter equal to $1000$ cubic centimeters.
1. Hint 1
  A cubic meter is one meter long, one meter wide and one meter high, so it hides three lengths.
2. Hint 2
  Convert the volume to cubic centimeters, then to liters, before comparing it with one bottle.
Answer

$32$ bottles.

Full solution

A cubic meter hides three lengths, so the meter-to-centimeter factor is used three times, which cubes it.
One cubic meter is $100^3=1{,}000{,}000$ cubic centimeters, so the tank holds
$0.048(1{,}000{,}000)=48{,}000$
cubic centimeters.

The factor of one liter per thousand cubic centimeters has cubic centimeters below, so they cancel, and the volume is
$48{,}000\div1000=48$
liters.

The number of bottles is
$48\div1.5=32$

As a check, $32$ bottles of $1.5$ liters hold $48$ liters, which is $48{,}000$ cubic centimeters, the tank's volume.

Answer

$32$ bottles.

Key idea

A cubed length factor converts the volume, and further true factors can then carry it to the unit the containers use.
Problem 6: Mark complete
Problem 7 The board game units

A board game measures lengths in two made-up units, the track and the step. Its rulebook says that $5$ square tracks cover the same area as $320$ square steps. How many steps long is one track?
1. Hint 1
  A square track is one track long and one track wide, so it hides two lengths.
2. Hint 2
  First find how many square steps make up one square track.
3. Hint 3
  Then find the positive number whose square is that count of square steps.
Answer

$8$ steps per track: one track is $8$ steps long.

Full solution

Five square tracks equal $320$ square steps, so one square track equals
$\frac{320}{5}=64$
square steps.

A square track is one track by one track, so converting it to square steps uses the length factor twice, which squares it.
The number of steps in one track must therefore have a square of $64$ .

Since $8^2=64$ and a length is positive, one track is $8$ steps long.

As a check, one square track is $8\cdot8=64$ square steps, so five square tracks are $5\cdot64=320$ square steps, as the rulebook says.

Answer

$8$ steps per track: one track is $8$ steps long.

Key idea

An area conversion uses the square of the length conversion, so a known area relationship reveals the length relationship.
Problem 7: Mark complete
Problem 8 What canceling does not prove

Nina converts a $3.5$ kilogram bag of flour to grams. She multiplies by $\frac{100\text{ g}}{1\text{ kg}}$ , the kilograms cancel, and her result of $350$ carries grams, so she says the canceling proves her conversion is right. Explain what the canceling does show and what it does not, state the test a fraction must pass before it may be used as a conversion factor, and give the bag's mass in grams. Use $1$ kilogram equal to $1000$ grams.
1. Hint 1
  Multiplying by a factor leaves an amount unchanged only when the factor itself equals one, and canceling alone does not arrange that.
2. Hint 2
  Test her fraction the way the lesson tests any factor: do its top and bottom name the same amount?
3. Hint 3
  Rebuild the factor from the given equality between kilograms and grams, then multiply the mass by it.
Answer

Canceling shows only that the factor is correctly oriented, not that it is true. A fraction may be used only when its top and bottom name equal amounts, and $100$ grams is not $1$ kilogram, so Nina's may not. The bag is $3500$ grams.

Full solution

Canceling tracks the units alone.
The kilograms attached to $3.5$ kg meet the kilograms below Nina's fraction and disappear, so grams survive, which shows only that her fraction is correctly oriented.

A conversion factor must also be true: its top and bottom must name the same amount, so that the fraction itself equals one.
Hers do not, since $100$ grams is a tenth of a kilogram, so her fraction equals one tenth.

Multiplying by one tenth shrinks the mass instead of renaming it, which is why her $350$ grams is a tenth of the true mass.

Built from the given equality, the true factor is $\frac{1000\text{ g}}{1\text{ kg}}$ , whose top and bottom are the same mass.
It gives
$3.5(1000)=3500$
grams.

As a check, $3500$ grams is $3500\div1000=3.5$ kilograms, the mass Nina started with, so only the unit changed.

Answer

Canceling shows only that the factor is correctly oriented, not that it is true. A fraction may be used only when its top and bottom name equal amounts, and $100$ grams is not $1$ kilogram, so Nina's may not. The bag is $3500$ grams.

Key idea

Reaching the target unit shows only that a factor is correctly oriented; the amount survives only when the factor's top and bottom name equal amounts, so that it equals one.
Problem 8: Mark complete
Problem 9 The printed capacity

A label converts cubic yards to cubic feet by multiplying the numerical measure by $27$ . Jo says converting cubic feet back to cubic yards also multiplies by $27$ , since both conversions concern volume. Is Jo correct? Give a positive-volume example to justify your decision. Use $1$ yard equal to $3$ feet.
1. Hint 1
  The direction of conversion matters as well as the power.
2. Hint 2
  A reverse volume conversion uses the cube of the reciprocal length factor.
Answer

No; divide by $27$ . For example, $81$ cubic feet is $3$ cubic yards.

Full solution

One yard is three feet, so one cubic yard is $3^3=27$ cubic feet.
Returning from cubic feet to cubic yards therefore uses numerical factor $1/27$ .

For example,
$81\cdot\frac1{27}=3$
Thus $81$ cubic feet is $3$ cubic yards.
Converting forward checks as $3(27)=81$ cubic feet.

Jo's rule would give $81\cdot27=2187$ cubic yards, not $3$ , so the rule is false.

Answer

No; divide by $27$ . For example, $81$ cubic feet is $3$ cubic yards.

Key idea

Converting a cubed unit back uses the reciprocal of the whole cubed factor.
Problem 9: Mark complete
Problem 10 The speed setup

Dev converts $36$ kilometers per hour to meters per second. He multiplies the speed by the factor $\frac{1000\text{ m}}{1\text{ km}}$ and then by the factor $\frac{3600\text{ s}}{1\text{ h}}$ . Explain what is wrong with his setup, and give the correct speed in meters per second. Use $1$ kilometer equal to $1000$ meters and $1$ hour equal to $3600$ seconds.
1. Hint 1
  Follow each unit through the setup: an unwanted unit cancels only when it appears once on top and once on the bottom.
2. Hint 2
  Check each factor in turn: which unit is it meant to remove, and where does that unit sit in the speed?
3. Hint 3
  Once every unwanted unit cancels, multiply the numbers on top and divide by the numbers on the bottom.
Answer

His time factor $\frac{3600\text{ s}}{1\text{ h}}$ is upside down; it should be $\frac{1\text{ h}}{3600\text{ s}}$ , and the correct speed is $10$ meters per second.

Full solution

Write the speed as $\frac{36\text{ km}}{1\text{ h}}$ .
Kilometers sit on its top and on the bottom of $\frac{1000\text{ m}}{1\text{ km}}$ , so they cancel, and that factor is oriented correctly.

Hours sit on the bottom of the speed and also on the bottom of $\frac{3600\text{ s}}{1\text{ h}}$ , so they do not cancel.
His product is left with meters times seconds on top and hours squared below, $\text{m}\cdot\text{s}/\text{h}^2$ , which is not meters per second.
So the time factor is upside down.

Flipping it to $\frac{1\text{ h}}{3600\text{ s}}$ puts hours on top, so they cancel and leave meters over seconds.
The correct speed is
$\frac{36(1000)}{3600}=10$
meters per second.

As a check, $10$ meters per second for the $3600$ seconds of one hour covers $36{,}000$ meters, which is $36$ kilometers.

Answer

His time factor $\frac{3600\text{ s}}{1\text{ h}}$ is upside down; it should be $\frac{1\text{ h}}{3600\text{ s}}$ , and the correct speed is $10$ meters per second.

Key idea

Following the units exposes an upside-down factor: the unit it should remove survives instead of canceling.
Problem 10: Mark complete

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