Lesson 7 · from §87 of the 1910 text
Problems with Letters
Problems with letters
In these problems some of the given numbers are letters. Solve them just as if the letters were numbers; the answer is an expression in those letters.
Example. If one pen costs p cents, n pens cost np cents. If m pens cost k cents, one pen costs \frac km cents.
Example. The sum of two numbers is s and the first is twice the second. Let x be the first and y the second. Then x + y = s and x = 2y. Substituting, 3y = s, so y = \dfrac s3 and x = \dfrac{2s}{3}.
A useful check: put numbers for the letters. With s = 12, the numbers are 8 and 4. ✓
The area of a triangle is half the product of its base and altitude: A = \frac12 bh.
Try one:
Quick check
The base of a triangle is 2a and the altitude is b. Find the area.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Solve each problem. The answers are expressions in the letters.
If one book costs a dollars, what will c books cost?
If a books cost b dollars, what will one book cost? c books?
(a) Find the perimeter and the area of a rectangle whose length is a and whose width is b. (b) Then find the perimeter and the area of a second rectangle whose dimensions are three times the first. (c) The perimeter of the second is how many times the perimeter of the first? (d) The area of the second is how many times the area of the first?
The base of a triangle is 8. The altitude is 10. Find the area.
The base of a triangle is b. The altitude is 8. Find the area.
The base of a triangle is b. The altitude is a. Find the area.
The base of a triangle is a + b. The altitude is a - b. Find the area.
The base of a triangle is x - 2y. The altitude is x + 2y. Find the area.
The area of a triangle is k. The base is b. Find the altitude.
The altitude of a triangle is a inches and the base is 10 inches. If 2 inches are taken from the altitude, how much must the base be increased so that the area will be the same as before?
The altitude of a triangle is a feet, the base is b feet. The altitude is increased h feet and the base decreased so that the area is the same as before. How many feet are taken from the base?
The sum of two numbers is s and their difference is d. Find the numbers.
The first of two numbers is a times the second, and the first minus the second is b. Find the numbers.
The sum of two numbers is b, and the quotient of the first divided by the second is a. Find the numbers.
If a is added to the numerator of a certain fraction, the value of the resulting fraction is 2. If b is added to the denominator, the value of the resulting fraction is 1. Find the fraction.
If the numerator of a certain fraction is increased by 1, the value of the resulting fraction is x. If the denominator of the fraction is decreased by 2, the value of the resulting fraction is y. Find the numerator and the denominator.
The value of a certain fraction is b. If 2 is added to the numerator, the value of the resulting fraction is c. Find the numerator and the denominator.
A boy who weighs a pounds and one who weighs b pounds balance at the opposite ends of a teeter board whose length is l feet. How far is the fulcrum from each end of the board?
A certain number of books at 80 cents each and another number at $1.10 each cost together h dollars. If the prices of the books had been interchanged, the total cost would have been k dollars. Find the number of each kind.
Two books cost c dollars. The first cost d cents more than the second. Find the cost of each in cents.
A and B have together k dollars. A gives h dollars to B and then they have equal sums. How many dollars had each at first?
If A gives h dollars to B, they will have equal sums. If B gives k dollars to A, A will have twice as much as B. How many dollars has each?
If A gives $10 to B, B will have h dollars more than A. But if B gives k dollars to A, A will have three times as much as B. How many dollars has each?