Lesson 8 · from §74 of the 1910 text
Problems: Numbers, Ages, Money, and Areas
More problems
These problems use everything from the chapter. Read carefully, name the unknown, and write the equation the words describe.
Quotient and remainder
When one whole number is divided by another, \frac{\text{Dividend}}{\text{Divisor}} = \text{Partial quotient} + \frac{\text{Remainder}}{\text{Divisor}}.
Example. Find two numbers whose sum is 95, such that the greater divided by the less gives a partial quotient of 4 and a remainder of 5.
Let x = the smaller number. Then 95 - x = the greater, and \frac{95 - x}{x} = 4 + \frac5x. Multiplying by x: 95 - x = 4x + 5, so x = 18 and 95 - x = 77. Check: 77 \div 18 = 4 with remainder 5. ✓
Interest
Money at 6% earns .06 of itself each year. If x dollars of $3100 are at 6%, the rest, 3100 - x, is at the other rate.
Coins
Count value in cents: n quarters are worth 25n cents.
Areas
Write the area before and after the change, using the same letter for the width in both.
Try one:
Quick check
Separate 50 into two parts such that their quotient is 4.
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.
Write an equation, solve it, and check the answer against the words of the problem.
Separate 300 into two parts such that their quotient is 5.
Separate 60 into two parts such that \frac23 of the greater will equal \frac34 of the smaller.
Separate 45 into two parts such that the sum of \frac49 of the greater and \frac23 of the smaller will be 24.
Separate \frac35 into two parts such that \frac13 of one part will equal \frac34 of the other.
Find two numbers whose sum is 95, such that the greater divided by the less gives a partial quotient of 4 and a remainder of 5.
Separate 126 into two parts such that one divided by the other gives a partial quotient of 6 and a remainder of 7.
The sum of two numbers is 1906. The greater divided by the less gives a partial quotient of 41 and a remainder of 16. Find the numbers.
Separate \frac85 into two parts such that their product is less by \frac25 than the square of the greater part.
Separate 71 into two parts such that 40 exceeds \frac23 of one part as much as the other exceeds 16.
A boy's age is now \frac25 of what it will be 12 years hence. How old is he?
Two thirds of a man's age now, equals \frac65 of what it was 30 years ago. Find his present age.
One sixth of a man's age 8 years ago equals \frac18 of his age 12 years hence. What is his age now?
A man invests part of $3100 at 6% and the remainder at 5%. The 6% investment yields annually $18.60 less than the 5% investment. Find the sum invested at 5%.
A man invests part of $5360 at 5% and the remainder at 6%. The yearly income from the 5% investment is $63.40 more than that from the 6% investment. Find the sum invested at 6%.
A part of $3880 is invested at 4% and the remainder at 6%. The total yearly income is $171.20. Find the amount invested at 6%.
A collection of five-cent pieces and quarters contains 80 coins. Their total value is $16. How many are there of each?
Twenty-eight coins, dimes and quarters, have the value of $5.05. How many are there of each?
The square of half a certain even number is 11 less than \frac14 the product of the next two consecutive even numbers. Find the numbers.
The square of \frac43 of a certain even number is 2864 less than 1\frac79 times the product of the next two consecutive odd numbers. Find the number.
A rectangle is four times as long as it is wide. If it were 4 meters shorter and 1\frac12 meters wider, its area would be 11 square meters more. Find its length and its breadth.
The length of a certain rectangle is 2\frac12 times its width. If it were 5 meters longer and 4 meters narrower, its area would be 50 square meters less. Find its dimensions.
It costs as much to sod a square piece of ground at 20 cents per square meter as to fence it at 80 cents per meter. Find the side of the square.
A rectangular court is twice as long as it is wide. It costs as much to fence it at 50 cents per yard as to sod it at 15 cents per square yard. Find its dimensions.
A rectangular picture 2\frac12 times as long as wide is surrounded by a frame 2 inches wide. The area of the frame is 128 square inches. Find the dimensions of the picture.
A square court has the same area as a rectangular court whose length is 2\frac79 yards greater and whose width is 2\frac12 yards less. Find the dimensions and area of each.
A man bought apples at 18 cents per dozen. He sold \frac15 of them at the rate of 3 for 4 cents, and the remainder at the rate of 4 for 3 cents, losing 76 cents. How many did he buy?