Project 8: Solving Equations, Part III (Word Problems) Translating between words and math: Common words and the math symbols they represent Expression part Words indicating this operation Order Addition plus, sum, total, altogether, and, Order doesn’t matter for addition. add(ed), increase by, more than Subtraction minus, subtract from, difference ASK: Which quantity are you starting with, and which quantity are you taking away from that between/of, less than, decreased/reduced by, take starting value? away from  minus ↔  decreased/reduced by ↔ Whatever comes after “by” is what is being taken away, even if other words have a different order.  difference between/of and ↔  subtract/take away from ↔ Whatever comes after “from” is the starting value, even if other words have a different order.  less than ↔ Multiplication multiply, product, times, of, Order doesn’t matter for multiplication. twice/doubled (times 2) Division divide(d) by, quotient/ratio of, ASK: Which quantity are you starting with, and per, go(es) into which quantity is dividing up the original value?  divided by ↔ or Whatever comes after “by” is what is doing the dividing, even if other words have a different order.  per ↔ or  Quotient/ratio of and ↔ or goes into ↔ or Whatever comes after “into” is the starting value, even if other words have a different order. Exponents Equals sign Variable Subtraction/Division Practice: Sentence The difference between four and a number The quotient of six and a number A number decreased by two Six goes into a number Eight miles per two gallons Five reduced by a number Take five away from a number From six, take away a number Divide three into a number Divide a number by three A number minus seven Take three away from a number From three, take away a number The difference between six and two squared, cubed, to the power of is/are/was/were, yields a number Can pick any letter—it’s up to you! Which is the original quantity? four Write using math symbols 4
Multiple operations at once and parentheses: Sentence Three times five, minus four Three times the difference between five and four The product of nine more than a number and two The product of two and nine, plus a number The difference of two and a number, times seven The difference of a number times seven and two Does the first operation apply to the second quantity mentioned, or the result of the second operation? Second quantity (5) Result (difference) 4 or 3 ⋅ 5
3⋅ 5 4
9
4 ⋅2
Note that “the product of nine” doesn’t make sense on its own—a product needs two things to be multiplied! Second quantity (9) 2⋅9
Result (a number times seven) or 2 ⋅ 9
2
Second quantity (a number) ⋅7
The sum of four times a number and two The sum of four and two, times a number Twice the difference of five and a number The difference of twice five and a number Five less than a number divided by five Five less than a number, divided by five A number subtracted from three, times four A number subtracted from three times four Six times the difference of a number and two The difference of six times a number and two Three times the square of a number Subtract two from the cube of a number ⋅7
2 or ⋅ 7
Note that “the difference of a number” doesn’t make sense on its own—a difference needs two things: the original value and the amount being taken away! Result (three times a number) Second quantity (three) 3⋅5
Result (nine more than a number) Eight less than three times a number Eight less than three, times a number The ratio of nine and a number, minus five The ratio of nine less than a number and five Six less than the total of a number and five Six less than a number, plus five Write using math symbols 3
3
8 8
2 Percentage increase/decrease word problems Many real‐life problems involve percentage increase/decrease. These problems include three values, one of which will be the unknown (and therefore represented by a variable): 


The original value/quantity, The percentage increase/decrease in decimal form, The new value/quantity, or ⋅
. The relationship between these three values is always: ⋅
This makes sense, because we multiply the original value by the percentage (in decimal form) to get the amount by which the original value increases/decreases, and we have to add that to the original value to get the new value. So ⋅ represents the amount by which the original value is changing. The percentage increase/decrease ( ) will be negative if the original value decreases and positive if the original value increases. Identify , , Solve the equation Which is unknown? Write the equation. 1
0.30
$105
A pair of shoes are on sale O=unknown
P=
—
0.30
0.7
$105
for 30% off. The sale price .
$
N=$105 is $105. What was the .
.
original price of the shoes? ⋅ 1 $150
1
$150
1
0.30
$105
Original problem Check your answer for reasonableness The original price ($150)should be
larger than the new price ($105).
The change in price seems reasonable,
given the size of the percentage (0.30
is about one third, and the decrease
from $150 to $105 of $45 is about
one third of $150.
O=$550
The new rent ($563.75)should be
$550 1 0.025
Your apartment rent P=0.025
larger than the original rent ($550).
$550 1.025
increases by 2.5% next $563.75
N=unknown The change in rent seems reasonable,
month. Your currently given the size of the percentage (2.5%
monthly rent payment is $550 per month. What will 1
is a quarter of 10%, and a 10%
your new monthly rent increase would be $55, so the size of
$550 1 0.025
payment be next month? this increase seems about right.
This is a decrease, so the percentage
2500 1
2400
Enrollments at school went O=2500
P=unknown
should be negative. And a 4%
2500 ⋅ 1 2500 ⋅
2400
from 2500 students to 2500 2500
2400
decrease is a bit less than half of
2400 students. What was N=2400 2500
2500
2500
10%. Since 10% of 2500 would be
the percentage decrease in 2500
2400
enrollments? 250, it seems reasonable for a
1
2500 2500 2500
100 decrease of 100 to be about 4%.
2500 1
2400
0 2500
100
2500
100
1
1) A restaurant bill is $25 before tip. You plan to leave an 18% tip in addition. How much should you pay in total? 2) The cost of a shirt went from $60 to $45 during a sale. What was the percent reduction in 0.04
0.04
4%
price? 3) Today there are 54 bears in a nature preserve. This is 20% more than there were last year. How many bears were there last year? 4) The number of students participating in sports went up from 250 to 280. What is the percent increase? 5) You paid $225 for a TV, including tax. Tax is 12.5%. What was the original price of the TV? 6) A store is currently selling backpacks for $75 each. They plan to reduce the price by 15%. What will the resulting sale price of a backpack be? Proportion problems For proportion problems, we are given two numbers that have a specific relationship to one another, and then are given a third number and asked which number would have the same relationship to this number as the other two numbers have to one another. So for a proportion problem there are always four values, one of which is unknown: 


Two values that have a relationship to one another that is known One additional number that is known An unknown that is supposed to have the same relationship to the For proportion problems, we will always set up these four values in the following way: The key point is that each vertical pair (
belong together in the same way. ,
,
and each horizontal pair (
,
,
) should Here is an example to illustrate this: A recipe uses 3 cups of flour to make 9 scones. If you want to make the same recipe for only 6 scones, how many cups of flour should be in the recipe?  9 and 6 both represent scones, so they are a pair  3 cups makes 9 scones, so these are a pair The unknown is the number of cups that goes along with 6 scones. Let’s call this . There are several possible equations we could write to describe the relationships given in this problem: 1)
2)
3)
4)
Each of these equations is equally good, and all of them will produce the same answer when we solve for . They all work because in each equation, the 9 and 6 are lined up (either vertically or horizontally), and the 9 and 3 are lined up (either vertically or horizontally). Which one should we pick? Any of them are fine, but it will be less work to solve the equation if we pick one of the equations with the variable in the top instead of the bottom. So let’s go with this one: ⋅
⋅ ⋅ 2
⋅ 1 2
2 So 2 cups of flour should be needed if the recipe is reduced in order to make only 6 scones. This makes sense because 2 cups is fewer than the 3 cups needed to make 9 scones (and 9 is bigger than 6). Now you try! On the next page, solve the provided proportion problems, using the problem above and the sample problem on the next page as an example. Original problem Identify which pairs of values belong together Which is unknown? Pick an equation and write it. If 5 sticks of butter  5 and 2 both represent sticks of butter, so weighs 565 grams, they are a pair how much do 2  5 sticks and 565 grams both represent the sticks of butter same amount of butter, so they are a pair weigh? The unknown is the number of grams that correspond to 2 sticks of butter, or . Equation: 1)If three sandwiches cost $16.50, how much do five sandwiches cost? 2)If you can run 3 miles in 24 minutes, how many miles at that same speed can you run in 20 minutes? 3)A recipe calls for ½ tsp ginger for every 3 cups of flour. How many tsp of ginger do you need to make the same recipe if you only use 2 cups of flour? 4)There are 6 teachers for every 75 students. To keep the same ratio, how many teachers are needed for 25 students? 5) There is a speed limit sign for every 1.75 miles of road. For 14 miles of road, how many speed limit signs should there be? Solve the equation Check your answer for reasonableness ⋅
⋅ 113 ⋅ 2
⋅ 226
⋅ 1 226
226 grams The number of grams that correspond to 2 sticks of butter is less than the number of grams that correspond to 5 sticks of butter. Also, 2 sticks of butter is a bit less than half of 5 sticks, so it makes sense that 2 sticks would correspond to 226 grams, which is a bit less than half of 565. Now practice writing different kinds of equations and solving them: First identify if this is a percentage Solve that equation increase/decrease problem, a proportion problem, or another type of word problem. Then, translate the words into an equation. 1) A hospital has an average of 2 nurses for every 5 patients. To maintain this ratio with 35 patients, how many nurses are needed? 2) Two more than the product of 3 and a number is 4. What is the number? 3) Your salary was $42,000, but you received a 3.5% raise. What is your new salary? 4) Eight is six less than two times a number. What is that number? 5) Twenty subtracted from five times a number is ten. What is the number? 6) Two thirds a cup of milk are needed for a muffin recipe that yields 24 muffins. How many cups of milk are needed to make 12 muffins using the same recipe? 7) Fifteen is three times the difference between a number and four. Find the number. 8) You negotiated a 5% decrease in rent. The rent you agreed to pay is $570. What was the original rent for the apartment? 9) A car traveled 24 miles every 30 minutes. At that speed, how many miles will the car travel in 10 minutes? 10) Last week you got a 75 on the test. This week, you got a 90. What is the percent increase in your test score? 11) A country needs 45 workers for every 10 retirees to pay retirees’ social security. How many workers would be needed to support 6 retirees under this system? Check the solution