8) Unit conversions & percent word problems

Question 1

What must you know in order to be able to successfully convert numbers from one unit to another?

Answer 1

Conversion factor
> statement of equality between two units
> therefore, can be expressed as a FRACTION with a value of 1

e.g., $1 = 100 cents

100 cents / 1 dollar = 1

Helpful tips:
> add unit signs to make sure the right units are being cancelled
> if you are asked about 1 UNIT of something, you need to make sure that unit is in the DENOMINATOR
e.g., If 200 gallons equals 76000 cm^3, how many cm^3 are in 1 pint (given that 1 pint = 8 gallons) —> NEED DENOMINATOR TO BE 1 pint

Question 2

Conversion involving two sets of units

Answer 2

When converting two units, convert one unit at a time
> just make sure if you include units in your conversion that they cancel out appropriately

Question 3

Unit conversions involving squared and cubed units

e.g., what is 81 inches^2 in feet^2? (note: 1 foot = 12 inches)

Answer 3

Need to make sure the conversion factor has the same units as what you want

*** > FIRST calculate the actual conversion factor by raising BOTH sides of the original unit conversion

e.g., 1 ft = 12 inches, we want inches^2

(1ft)^2 = (12 inches)^2
1 ft^2 = 144 inches^2

Therefore, once we calculate inch^2, we convert to feet^2
= 81 inches^2 * 1 ft^2 / 144 inches^2

= 81 / 144 feet^2
= 9/16 feet^2

Question 4

What should you remember for hard conversion questions (involving 3+ conversion factors)

e.g., a certain ice cream shop uses 10 tablespoons of ice cream per customer. If the ice cream shop purchases its ice cream in 3-gallon containers, how many containers will it need to purchase for a 6-day supply, if customers come in the store at the constant rate of 64 customers per day (1 gallon = 256 tablespoons)?

Answer 4

10 tablespoons of ice cream = 1 customer

1 container = 3 gallons of ice cream

64 customers = 1 day

1 gallon = 256 tablespoons

For these tricky conversions, figure out the FINAL UNITS you need and then work backwards (but first deal with integer constraints…)

In this case, we want to know # of containers, so we need to know # of gallons of ice cream, which we can figure out once we know # of tablespoons of ice cream for the 6 days

To get # of tablespoons of ice cream for the 6 days, we need to know # of customers over 6 days…

Ans 5 containers

Question 5

Helpful tips when solving percent word problems with an unknown total

Answer 5

Either set the unknown total equal to “100” or use a variable like “x”

Question 6

Wording: Percent of

E.g., What is 5 percent of z?

Answer 6

“Of” signals MULTIPLICATION

“Percent” signals “/100” or “divide by 100”

e.g., 5/100 * z

Question 7

What is the value of one one-thousandth percent of z?

Answer 7

Translate sentence carefully

“one-thousandth” = 0.00x (recall decimal places to the right are tenths, hundreths, one-thousandths, ten thousandths, hundred thousandths, millionths….)
> means 1 part in 1000

“one one-thousandth” = 0.001

“percent” = /100

so one one-thousandth percent of z = 0.001/100 = z/100000

Question 8

Wording: “what percent”

E.g., 100 is what percent of 50?

What percent of 50 is 100?

Answer 8

“What” = “x”

Therefore, “What percent” = x/100

e.g., 100 = (x/100)*50

e.g., x/100 * 50 = 100

Alternatively, you can express as part / total

e.g., 100/50 * 100 = x

Question 9

Wording: Percent less than

e.g., Value is reduced 20%

e.g., 30% less than $40 is what?

Answer 9

Not necessarily have “percent less than” in the question, but the effect is a REDUCTION in the value of some asset e.g., discount the price of merchandise

Final value = (1 - x% reduced) * initial value

e.g., if value is reduced 20%, then the final value is 80% of the original value

e.g., 30% less than $40 is the same as $40 reduced 30%

Note:
> don’t need “by” such as “reduced by 30%)

Question 10

Wording: Percent greater than

e.g., price increased 30%

Answer 10

Refers to an INCREASE in value of something e.g., mark up the price of merchandise

Final value = initial value * (1 + x% increase)

Note:
> don’t need “by” such as “increased by 30%)

Question 11

Sometimes the ORDER of how you convert units makes the problem easier or harder - how do you know the right order of problem solving?

e.g., integer quantities

Example: Scott has ordered 25 packages of margarita mix for his party. Each package contains 5L of mix. If Scott needs a total of 1500 cups of mix for his party and each package of mix costs $3.00, how much more will Scott have to spend to get 1500 cups of mix? (Note: 1 liter = 4.2 cups)

Answer 11

FOR ANY unit conversion question (hint: see if there are conversion factors provided)…

(1) Figure out if there are any quantities that MUST BE INTEGERS
> this will mean that you might have to ROUND UP to produce an integer value

(2) Solve for the right value of integer metrics first

(3) Then work with different operations to get the other units to cross out in the way you want it to

Example:
> in this question, # of packages must be an INTEGER NUMBER –> figure out how many packages Scott needs first (based on gap in # of cups)

> Scott has 25 packages
he needs 1500 cups = 500/7 packages = 71.xxx packages = ROUND UP to 72 packages
therefore Scott needs to buy 72-25 = 47 more packages
now figure out the cost of the 47 additional packages
= 47 * $3
= $141

ALSO tip:
> Use your knowledge of FACTORS to help with simplification of fractions

e.g., (4320 * 10^3)/(36 * 24 * 100)

—> 36*24 = 864 –> a factor of 4320

Question 12

Common unit conversions Hours to seconds

Answer 12

1 hour = 3600 seconds = 60 mins / h * 60 sec / min

> use the above INSTEAD of manually writing out 60*60 each time

Question 13

Percent change formula

Answer 13

(P2/P1 - 1)*100

(P2-P1)/p1 * 100

Tells us the % increase (+ ans) or decrease (- ans) in two values
e.g., if % change = -20% —> this means the value DECREASE 20%

Note:
> Percent Change is related to “percent greater than” and “percent less than”

e.g., 1000 is what percent greater than 900?
> cannot do part / whole (“percent OF”)
> need to set up equations properly

Percent greater than: initial value * (1 + x/100) = final value
Percent less than: initial value * (1 - x/100) = final value

Question 14

Imbalance in percent changes

Answer 14

Rule 1: To return a discounted number back to its original value, we must increase the discounted value by some percent GREATER than the percent decrease by which it was discounted

e.g., 100 is reduced by 20% becomes 80. But 20% increase of 80 does not equal 100 (96 is less than 100) –> undershoots the increase

Rule 2: To return a marked-up number back to its original value, we must decrease the marked-up value by some percent LESS than the percent increase by which it was marked up

e.g., 100 is increased by 20% becomes 120. But 20% decrease of 120 does not equal 100 (96 is less than 100) –> overshoots the decrease

To figure out the exact % change needed to bring back to original number, use the percent change formula

Question 15

Hard percent wording question:

The price of an item was increased by 20 percent. By what fraction of the new price must the new price of the item be decreased to return it to its original price?

Answer 15

Pay attention to wording:
> We are looking for an answer that is the FRACTIONAL FORM of the % decrease

> NOT the FACTOR why which the new price must be multiplied by

ans: -1/6 —> 1/6

Question 16

% profit formula

Answer 16

(Profit / Cost) * 100
=(Price - Cost)/Cost
= positive or negative percentage (symbolizes positive or negative profit)

NOT profit margin (/revenue)

e.g., -20% profit means profit = -0.2 * cost

Question 17

Tricky PS percents wording:

A customer buys two products, X and y, and spends a total of $350. Both products come with rebate coupons. The rebate on the price of product X is 15 percent, and the rebate on the price of product Y is 10 percent. If x and y are the prices of products X and Y, respectively, before the rebate and the total rebate earned by the customer was $45, how much did each product cost?

Answer 17

Wording - you need to interpret that $350 refers to the PRE-REBATE cost

> then solve linear systems of equations

Ans x = $200 and y = $150

Question 18

Tricky percent DS problem

Breck owns a lawn moving business that has one employee. Breck earns $100 in revenue for each lawn that his employee mows, pays his employee y dollars for each lawn that the employee mows, and must pay z percent of the remaining revenue in tax. Assuming he has no other expenses, what percent of the revenue remains after Breck pays his employee and the tax?

(1) Breck pays his employee 20 percent of the total revenue earned per lawn

(2) If Breck did not have to pay an employee, but still paid z percent tax, he would pay $15 in tax per lawn

Answer 18

Don’t get confused by your messy handwriting! Stay focused

Ans C

Formula we are solving for is profit/revenue
= (100-y)*(100-z)/100 * (1/100)

(1) y = 20%*100 = 20 –> don’t know anything about z tax rate

NS

(2) 15 = z% * 100
z = 15 –> don’t know anything about y employee cost

*Don’t confuse the fact that this condition said “if y = 0, then would pay $15 in tax” –> we don’t know what y is!

(3) sufficient

Note:
> Don’t assume that the conditions presented in one of the statements is the same as for the broader question

Question 19

Tricky DS percents wording:

Karen purchased 10 shares of stock for x dollars per share. She later sold the 10 shares for y dollars per share, but paid z percent tax on her profit. After tax, what was the profit Karen made on the sale of her stock?

(1) y - x = 5000
(2) if she had paid 10 percent more tax, Karen’s profit would have been $30,000

Answer 19

For tricky percent DS word problems, try to clearly outline the equation and what is UNKNOWN
> as you receive more information, figure out what’s still unknown and what can be solvable

In this case, we want profit = 10(y - x)(1 - z/100)

(1) y - x = 5000
But we still don’t know anything about z

NS

(2) 10% more tax means tax rate goes up by 10 pp

(z + 10)/100

Therefore, 30000 = 10(y - x)(1 - (z+10)/100) —-> still don’t know y - x or z
(3 unknowns)

NS

(3) combined, we know y - x = 5000 AND CAN USE STATEMENT 2 to solve for z

then, we are sufficient

C

Note:
> Don’t assume that the conditions presented in one of the statements is the same as for the broader question