What is seven plus the quotient of a number and 5 is -12

Question

Seven plus the quotient of a number and 5 is -12

Answer 1

The quotient is the result of division.

7 + n/5 = -12

Remember your basic terms. Sum is the result of addition. Difference is the result of subtraction, and product is the result of multiplication.

Answer 2

Question is improperly worded and grammatically incorrect. If the desired result is the equation represented, the question should be. "Write 'seven plus the quotient of a number and 5 is −12' as an algebraic equation". If the desired result is the unknown number, the question should be "What is the number such that seven plus the quotient of the number and 5 is −12", in which case the first step is to write the representing algebraic equation..

In either case, I want to provide a strategy to answer the question that can be applied to similar problems, and not just this one..

The way to write a verbal expression as an algebraic expression is a translation from words to algebraic symbols, where, in this case, "is" directly translates as "=", "plus" directly translates as "+", and "a number" directly translates as "n" or some other variable letter of your choosing. The "quotient of one number and another number" needs to be reworded for a direct translation and is the same as saying "one number 'divided by' another number" where "divided by" translates into either "÷" or "/"

So

"Seven plus the quotient of a number and 5 is −12"

is the same as saying

"Seven plus the a number divided by 5 is −12"

which algebraically translates as

7 + n/5 = −12

So for this, and similar problems, one should know the verbal description of various algebraic symbols, then translate. You might have to re-word the problem verbiage, somewhat, for a direct translation to algebra to work.

If the question wants you to find out what the number is, then you have to apply principles of equality.

First Isolate the variable term, n/5, on one side of the "=" by subtracting that which is being added to it, 7, from both sides of the "=", subtraction being the opposite operation to addition, resulting in

n/5 = −19

(a variable term being a variable multiplied or divided by either a number or another variable).

THEN

Isolate the variable in the now isolated variable term by multiplying that which is dividing it, 5, from both sides of the "=", multiplying being the opposite operation to division, resulting in

n = −95.

Notice the principles:

Use the same opposite operations, with the same numbers (or expressions in more complicated equations), to both sides of the "=" to isolate a variable term (or expression in more complicated equations, one level at a time), and, ultimately isolating the variable, by itself, on one side of the "=".

We then check our answer by substituting our result into the original equation to see if it works.

n/5 = −19

-95/5 = −19

−19 = −19

IT WORKS.

CAUTION 1:

Sometimes the temptation, in this problem

7 + n/5 = −12

would be to first multiply both sides by the dividing number, 5.

If you did this, you would have to remember to multiply both sides COMPLETELY by the 5.

5(7 + n/5) = 5(−12)

5(7) + 5(n/5) = −60, using the distributive law.

35 + n = −60

n = −60−35

n = −95

CAUTION 2

In general, when applying operations to both sides of an "=" of a number (or algebraic expression in more complicated advanced equations), one should avoid dividing by a variable, variable term or other variable expression, as this could eliminate possible solution(s). In the reverse, if one multiplies both sides of an "=" by a variable, variable term or other variable expression, you might introduce a solution(s) that may not work; thus the need to substitute each of your candidate answers back into the original equation to verify which of the solutions, if any, actually work.

CAUTION 3:

In more complicated advanced equations, it is possible that for the solution(s) you come up with, after doing everything correctly, none will work when substituting back into the original equation. In this case the answer is "null", or "no solution".

DM