How to Solve Word Problems Using , "Let Statements"
These problems usually appear as something like the following statement;
Tom is three times older than Harry. Five years from now their combined ages will be 70. How old is Tom and how old is Harry?
In order to solve this we need to form algebraic expressions from the statements in the word problem, these statements will take the form of, "let," statements.
Since Harry's age is what we know the least about, his current age will serve as the base for our first let statement and we will say;
Let Harry's age=X
Further from the first sentence we know that Tom is three times older than Harry, so we can form another let statement;
Let Tom's age=3X
We further know that 5 years hence the sum of their ages will be 70 So,
Let Harry's age 5 years from now=X+5, and
Let Tom's age 5 years from now =3X+5
The problem tells us that the sum of their ages 5 year from now will equal 70, so;
X+5+3X+5=70, and now we solve for X. Start by combining like terms,
4X+10=70, subtract 10 from both sides
4X=60, divide both sides by 4
X=15, now that we have found X we can go back to our original let statements to find their current ages
Harry's age=X, so Harry's age=15
Tom's age=3X, so Tom's age=3(15)=45
Check Your Work
We will check our work by plugging the value we have determined for X into the equation we solved;
70=70, so it would seem we solved the equation correctly, but to be thorough we need to go back and be sure that the let statements that led us to our equation make sense in terms of the word problem.
This requires another careful look at each let statement to determine that what we did with our original variable X, was correct. if you recall X was Harry's age, and a double check of the problem confirms that Tom's age is three times Harry's age so 3X is Tom's age, so far so good.
Now we must look at the expressions of age for 5 years hence, we should have added five to each expression, which we did, added them together for a combined age of 70. Does our equation reflect the word problem?
X+5+3X+5=70, yes it does.
It is important to double check not only the solving of this equation but the route you took to establish it, this is where careless errors are often made.
A More Difficult Problem
A women saves her monthly pay for a year. She then spends 1/2 of her money on her mortgage, then 1/3 of the remaining money on car insurance, and then 1/4 of the remaining money on clothes. After she bought all of that, she had $7777 left. Assuming her only source of income is her monthly paycheck, how much money does she earn every month?
Start with what we are look for, monthly pay and let that equal X
Now she saved her pay for a year before she began her spending spree so we need an expression for how much money she started with,
Let saved money=12X
Now we need expressions to represent her spending, she spent half on her mortgage so
Let money on mortgage=1/2(12X) She then spent a third of what remained on car insurance so,
Let money on insurance=(1/3)(1/2)(12X) She then spent a quarter of what remained on clothes so,
Let money on clothes=(1/4)(1/2)(1/3)(12x) Now let's simplify these expressions before we go on, so
Now we have an expression for the money she started with (12X), expressions for what she spent on her mortgage(6X), her car insurance(2X), and clothes(1/2X), and we know how much money she had after spending that money(7,777) so we can set up an equation and solve for X.
Start with the expression for her year's worth of saving subtract the expression for what she spent on her mortgage, subtract the expression for what she spent on car insurance, subtract the expression for what she spent on clothes and set it equal to 7,777. It should look like this;
12X-6X-2X-1/2X=7,777, combine like terms
3.5X=7,777, divide both sides by 3.5 so,
Now the question asked for how much she makes each month and since we let X=her monthly earnings, this is our answer, $2,222/month.
Be careful sometime the question to be answered is something like, how much did she save in a year? In other words not simply what you get for X but rather an expression containing X, if this was the case here then we would use the let statement, saved money=12X or $26,664.
Let's plug it back into our expression to see if it works
26,664-13332-4,444-1,111=7,777, crunch the numbers and
Again I recommend going back and making sure that the equation you wrote reflects what is said in the word problem. Make sure all let statement make sense and that you didn't make any mistakes in the multiplying together of fractions.