One number is 3 less than two times the other. If their sum is increased by 7, the result is 37. Find the numbers.

solution to Example 1 of Algebra Word Problems :

Let x be the other number. Then the first number as per data is 3 less than two times x. That means the first number is 2x - 3. Their sum = x + (2x -3) = 3x - 3. By data, Increasing their sum by 7, the result is 37. That means (3x - 3) + 7 = 37. ⇒ 3x - 3 + 7 = 37 ⇒ 3x + 4 = 37

This is the Linear Equation formed by converting the given word statements to the symbolic language.

Now we have to solve this equation. ⇒ 3x = 37 - 4 = 33 ⇒ x = 33⁄3 = 11. The first number = 2x - 3 = 2(11) - 3 = 22 - 3 = 19. The numbers are 11 and 19. Ans.

Check: Since 19 = 2(11) - 3, "one number is 3 less than two times the other" is satisfied. (verified) Since 19 + 11 + 7 = 37, "If their sum is increased by 7, the result is 37" is satisfied. (verified)

Example 2 of Algebra Word Problems

If two thirds, one half and one seventh of a number is added to itself, the result is 37. Find the number.

solution to Example 2 of Algebra Word Problems :

Let x be the required number. Then two thirds of the number = 2x⁄3. one half of the number = x⁄2. one seventh of the number = x⁄7. As per data, sum of these added to the number i.e.x = 37. That means 2x⁄3 + x⁄2 + x⁄7 + x = 37.

This is the Linear Equation formed by converting the given word statements to the symbolic language.

Now we have to solve this equation. The L.C.M. of the denominators 3, 2, 7 is 3 x 2 x 7 = 42. Multiplying both sides of the equation by 42, we get 42(2x⁄3) + 42(x⁄2) + 42(x⁄7) + 42x = 42 x 37. 14(2x) + 21(x) + 6(x) + 42x = 42 x 37 ⇒ (28 + 21 + 6 + 42)x = 1554. ⇒ 97x = 1554 ⇒ x = 1554⁄97 = 16.02 The required number = 1554⁄97 = 16.02 Ans.

Check: two thirds of the number = (2⁄3.)(1554⁄97) = (2)(518⁄97) = 1036⁄97 one half of the number = (1⁄2.)(1554⁄97) = 777⁄97 one seventh of the number = (1⁄7)(1554⁄97) = 222⁄97 the number = 1554⁄97 Sum of all these = (1036 + 777 + 222 + 1554)⁄97 = 3589⁄97 = (37 x 97)⁄97 ( Since 37 x 97 = 3589) = 37. Which is same as per data. (verified)

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