Lesson 2 · from §69 of the 1910 text
Problems Leading to Fractional Equations
Problems with fractions in them
Word problems often talk about "one fourth of a number" or "\frac25 of his age." Translate them exactly as before, then clear the fractions.
Example. The sum of two numbers is 38. One tenth of the greater equals \frac19 of the less. Find the numbers.
Let g = the greater number. Then 38 - g = the less, and \frac{g}{10} = \frac{38 - g}{9}. Multiply by 90: 9g = 380 - 10g, so 19g = 380 and g = 20. The numbers are 20 and 18.
Check: \frac{20}{10} = 2 and \frac{18}{9} = 2. ✓
Ages. If A is a years old now, then n years from now A is a + n, and n years ago A was a - n. "A's age is \frac52 B's age" becomes a = \frac52 b.
Sound and bullets. Time = distance ÷ rate. If a bullet flies d feet at 1925 feet per second, it is in the air \frac{d}{1925} seconds; the sound coming back takes \frac{d}{1100} seconds more.
Try one:
Quick check
One third of a number plus \frac15 of the number is 16. Find the number.
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.
One fourth of a certain number plus \frac1{12} of that number equals 16. Find the number.
The difference between \frac13 of a certain number and \frac1{17} of it is 70. Find the number.
The sum of two numbers is 38. One tenth of the greater number equals \frac19 of the less. Find the numbers.
The width of a rectangle is \frac45 of its length. The perimeter is 216 centimeters. Find the area of the rectangle.
What number must be added to the numerator of the fraction \frac47 so that the resulting fraction will be \frac15 of the number?
Three fourths of a certain number is \frac13 the sum of the next two consecutive numbers. Find the numbers.
A certain odd number divided by 11 is equal to \frac1{24} of the sum of the next two consecutive odd numbers. Find the numbers.
What number added to both terms of the fraction \frac{10}{17} gives a fraction whose value is \frac57?
Separate 42 into two parts such that \frac16 of their difference is \frac13.
One fourth the difference of three times a certain number and 4 equals \frac17 the difference of five times the number and 4. Find the number.
Separate 112 into two parts such that their quotient is \frac92.
There are two numbers whose sum is 24. If their difference be divided by their sum, the quotient will be 3\frac56 less than the difference of the two numbers. Find the numbers.
The quotient of 27 plus seven times a certain number, divided by twice the number, equals the quotient of 90 plus five times the number, divided by three times the number. Find the number.
A's age is \frac52 B's age. In 10 years A's age will be twice B's age. Find their ages now.
The age of A is \frac23 that of B. Fourteen years ago A's age was \frac12 B's age. Find their ages now.
A is 16 years older than B. Eight years ago B was \frac35 as old as A. Find their ages now.
Jupiter has 4 more moons than Uranus, and Saturn 2 more than twice as many as Uranus; Mars has 6 fewer than Jupiter, and Neptune half as many as Mars. These planets have together 25 moons. How many has each? (These were the counts known in 1910.)
A triangle has the same area as a trapezoid. The altitude of the triangle is 30 meters and its base is 8 meters. The altitude of the trapezoid is \frac13 that of the triangle, and one base equals the base of the triangle. Find the other base of the trapezoid.
A marksman hears the bullet strike the target 3 seconds after the report of his rifle. If the average velocity of the bullet is 1925 feet per second and the velocity of sound is 1100 feet per second, find the distance to the target and the length of time the bullet was in the air.
A gunner using one of the best modern rifles would hear the projectile strike the target 2640 yards distant in 9\frac25 seconds after the report of the gun, provided the projectile maintained throughout its flight the same velocity it had on leaving the gun. Find this velocity if sound travels 1100 feet per second.