Problems

494. 1. Divide $35 between two men so that their shares shall be in the ratio of 3 to 4.

2. Two numbers are in the ratio of 3 to 2. If each is increased by 4, the sums will be in the ratio of 4 to 3. What are the numbers?

3. Divide 16 into two parts such that their product is to the sum of their squares as 3 is to 10.

4. Divide 25 into two parts such that the greater increased by 1 is to the less decreased by 1 as 4 is to 1.

5. The sum of two numbers is 4, and the square of their sum is to the sum of their squares as 8 is to 5. What are the numbers?

6. Find a number that added to each of the numbers 1, 2, 4, and 7 will give four numbers in proportion.

7. In the state of Minnesota the ratio of native-born inhabitants to foreign-born recently was 5:2. What was the number of each, if the total population was 1,750,000 ?

8. A business worth $19,000 is owned by three partners. The share of one partner, $6000, is a mean proportional between the shares of the other two. Find the share of each.

9. What number must be added to each of the numbers 11, 17, 2, and 5 so that the sums shall be in proportion when taken in the order given?

10. Four numbers are in proportion; the difference between the first and the third is 23; the sum of the second and the third is 6; the third is to the fourth as 4:5. Find the numbers.

11. Prove that no four consecutive integers, as n, n + 1, n + 2, and n + 3, can form a proportion.

12. Prove that the ratio of an odd number to an even number, as 2m+1:2n, cannot be equal to the ratio of another even number to another odd number, as 2x: 2 y + 1.

13. The area of the right triangle shown in Fig. 1 may be

14. In Fig. 2, the perpendicular p, which is 20 feet long, is a mean proportional between a and b, the parts of the diameter, which is 50 feet long. Find the length of each part.

15. In Fig. 3, the tangent t is a mean proportional between the whole secant ce, and its external part e. Find the length of t, if e=93 and c = 50%.

16. The strings of a musical instrument produce sound by vibrating. The relation between the number of vibrations N and N' of two strings, different only in their lengths 7 and l', is expressed by the proportion

A c string and a g string, exactly alike except in length, vibrate 256 and 384 times per second, respectively. If the c string is 42 inches long, find the length of the g string.

17. If L and I are the lengths of two pendulums and T and t the times they take for an oscillation, then

T2: t2=L:l.

A pendulum that makes one oscillation per second is approximately 39.1 inches long. How often does a pendulum 156.4 inches long oscillate?

18. Using the proportion of exercise 17, find how many feet long a pendulum would have to be to oscillate once a minute.

VARIATION

495. Many problems and discussions in mathematics have to do with numbers some of which have values that are continually changing while others remain the same throughout the discussion. Numbers of the first kind are called variables; numbers of the second kind are called constants.

Thus, the distance of a moving train from a certain station is a variable, but the distance from one station to another is a constant.

Two variables may be so related that when one changes the other changes correspondingly.

496. One quantity or number is said to vary directly as another, or simply to vary as another, when the two depend upon each other in such a manner that if one is changed the other is changed in the same ratio.

Thus, if a man earns a certain sum per day, the amount of wages he earns varies as the number of days he works.

It is read varies as.'

497. The sign of variation is cc. Thus, xxy, read 'x varies as y', is a brief way of writing the proportion

x: x'=y: y',

in which x' is the value to which x is changed when y is changed to y'.

498. The expression xy means that if x is doubled, y is doubled, or if x is divided by a number, y is divided by the same number, etc.; that is, that the ratio of x to y is always the same, or constant.

x

If the constant ratio is represented by k, then when xxy,

-=

y

k, or xky. Hence,

If x varies as y, x is equal to y multiplied by a constant.

499. One quantity or number varies inversely as another when it varies as the reciprocal of the other.

Thus, the time required to do a certain piece of work varies inversely as the number of men employed. For, if it takes 10 men 4 days to do a piece of work, it will take 5 men 8 days, or 1 man 40 days, to do it.

If x varies inversely as y, their product is a constant.

500. One quantity or number varies jointly as two others when it varies as their product.

Thus, the amount of money a man earns varies jointly as the number of days he works and the sum he receives per day. For, if he should work three times as many days, and receive twice as many dollars per day, he would receive six times as much money.

In xyz, if the constant ratio of x to yz is k,

If x varies jointly as y and z, x is equal to their product multiplied by a constant.

501. One quantity or number varies directly as a second and inversely as a third when it varies jointly as the second and the reciprocal of the third.

Thus, the time required to dig a ditch varies directly as the length of the ditch and inversely as the number of men employed. For, if the ditch were 10 times as long and 5 times as many men were employed, it would take twice as long to dig it.

If x xaries directly as y and inversely as z, x is equal to

plied by a constant.

502. If x varies as y when z is constant, and x varies as z when y is constant, then x varies as yz when both y and z are variable.

Thus, the area of a triangle varies as the base when the altitude is constant; as the altitude when the base is constant; and as the product of the base and altitude when both vary.

PROOF

Since the variation of x depends upon the variations of y and z, suppose the latter variations to take place in succession, each in turn producing a corresponding variation in x.

While z remains constant, let y change to 1, thus causing x to change to x'.

Now while y keeps the value y1, let z change to z1, thus causing ' to change to x1.

Since, if both changes are made, x, y, and z, are constants, 1 is a constant, which may be represented by k.

Similarly, if x varies as each of three or more numbers, y, z, when the others are constant, when all vary x varies as their product.

v,

...

Thus, the volume of a rectangular solid varies as the length, if the width and thickness are constant; as the width, if the length and thickness are constant; as the thickness, if the length and width are constant; as the product of any two dimensions, if the other dimension is constant; and as the product of the three dimensions, if all vary.