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16 27 110

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By a due consideration and application of these Problems, many questions may be solved in a short and easy manner, although some of them are generally supposed to belong to higher rules.]

PROBLEM 1.

Having the sum of two numbers, and one of them given, to find the other.

Rule.-Subtract the given number from the given sum, and the remainder will be the sum required.

Let 288 be the sum of two numbers, one of which is 115, the other is required.

From 288 the sum,

Take 115 the given number:

Remains 173 the other.

PROB. 2. Having the greater of two numbers, and the dif ference between that and the less given, to find the less.

Rule. Subtract the one from the other.

Let the greater number be 325, and the difference between that and the other 198: What is the other?

From 325 the greater,
Take 198 the difference.

Remainder 127 the less.

PROB. 3. Having the lesser of two numbers given, and the difference between that and a greater, to find the greater.

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PROB. 4.

198 the difference.

Sum 325 the greater number required.

Having the sum and difference of two numbers given, to find those numbers.

Rule. To half the sum add half the difference, and the sum is the greater; and from half the sum take half the difference, and the remainder is the less.-Or, from the sum take the difference, and half the remainder is the least; to the least add the given difference, and the sum is the greatest.

What are those two numbers, whose sum is 48, and difference 14?

2)48

Half sum 24

2)11

Half difference=7

24+7-31 the greater; and 24-7-17 the least,

Or, 48-14-2=17; and 17+14=31.

PROB. 5. Having the sum of two numbers and the difference of their squares* given, to find those numbers.

Rule.-Divide the difference of their squares by the sum of the numbers, and the quotient will be their difference: You will then have their sum and difference, to find the numbers by Problem 4.

What two numbers are those, whose sum is 32, and the difference of whose squares is 256 ?

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PROB. 6. Having the difference of two numbers, and the difference of their squares given, to find those numbers.

* The square of a number is the product of it multiplied into itself.

H

Rule. Divide the difference of their squares by the difference of the numbers, and the quotient will be their sum; then proceed by Problem 4.

What are those two numbers, whose difference is 20, and the difference of whose squares is 2000?

20)2000(100 sum. 50+10=60, the greater; and 50-10-40 the less.

PROB. 7. Having the product of two numbers, and one of them given, to find the other.

Rule.-Divide the product by the given number, and the quotient will be the number required.

Let the product of two numbers be 288, and one of them 8; we demand the other.

8)288

Ans. 36

PROB. 8. Having the dividend and quotient, to find the divisor. Rule. Divide the dividend by the quotient.

Hence we get another method of proving division.

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PROB. 9. Having the divisor and the quotient given, to find the dividend.

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REMARK.-The scholar, having now surveyed the ground-work of his studies, begins to see their application to the common concerns of life;-and it is important, while proceeding in the higher rules, that his memory be strengthened by repeated examinations in the previous studies. The instructor is advised, therefore, to state questions of his own, promiscuously under the several rules, that the good scholar may have an opportunity of proving to his teacher and friends, by prompt and ready answers to difficult questions, that he thoroughly understands the subject before him.This hint, improved now, may be of essential service hereafter.

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