The sum of two numbers is 7 find when their product is greatest

We are trying to solve the system of equations $x+y=16$, $xy=55$. Here are a couple of systematic approaches that work in general.

Approach $1$: We will use the identity $(x+y)^2-4xy=(x-y)^2$. In our case, we have $(x+y)^2=256$, $4xy=220$, so $(x-y)^2=36$, giving $x-y=\pm 6$.

Using $x+y=16$, $x-y=6$, we get by adding that $2x=22$, and therefore $x=11$. It follows that $y=5$.

The possibility $x+y=16$, $x-y=-6$ gives nothing new. Adding, we get $2x=10$, so $x=5$, and therefore $y=11$.

Approach $2$: From $x+y=16$, we get $y=16-x$. Substitute for $y$ in $xy=55$. We get $x(16-x)=55$. Simplification gives $x^2-16x+55=0$. The quadratic factors as $(x-5)(x-11)$, so our equation becomes $(x-5)(x-11)=0$, which has the solutions $x=5$ and $x=11$.

But we cannot necessarily rely on there being such a straightforward factorization. So in general after we get to the stage $x^2-16x+55=0$, we would use the Quadratic Formula. We get $$x=\frac{16\pm\sqrt{(-16)^2-4(55)}}{2}.$$
Compute. We get the solutions $x=5$ and $x=11$. The corresponding $y$ are now easy to find from $x+y=16$.

Remarks: $1,$ Recall that the Quadratic Formula says that if $a\ne 0$, then the solutions of the quadratic equation $ax^2+bx+c=0$ are given by $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$

Your approach was along reasonable lines, but things went wrong in the details. From $xy=55$ we get $x=\frac{55}{y}$. Substituting in the formula $x+y=16$, we get $$\frac{55}{y}+y=16.$$ A reasonable strategy is to multiply through by $y$, getting $55+y^2=16y$, or equivalently $y^2-16y+55=0$. Now we have reached a quadratic equation which is basically the same as the one we reached above.

$2.$ The first approach that we used (presented as an algorithm, and stripped of algebraic notation) goes back to Neo-Babylonian times. The "standard" problem was to find the dimensions of a door, given its perimeter and area.

The 17th-century Indian mathematician Brahmagupta is the “father of arithmetic” and Carl Friedrich Gauss in 1801, provided the Fundamental principle of number theory. Arithmetic is elementary of mathematics that is majorly concerned with numerals,  number system, their properties, and their operations from ancient times. The term is derived from the Greek word “arithmos” which simply means numbers. The traditional methods of arithmetic operations are constituent of addition, difference, multiplication, and division. These operations are being carried out for the purpose of human social and economical development for centuries.

The branch of mathematics that specifically deals with the study of numbers and properties of traditional operations like addition, subtraction, multiplication, and division is called as Arithmetic.

Besides the traditional operations of addition, subtraction, multiplication, and division arithmetic also include advanced calculating methods for percentage, logarithm, exponentiation and square roots, etc.

Types of Basic Operation in Arithmetic

The four basic operations of arithmetic are discussed below:

Addition(+)

Addition or also known as summation is an operation to combine two or more values or numbers into a single value. The process of adding n numbers of value is called summation.

While adding 0 to any value gives the same result. Hence, 0 is known as the identity element of addition. For example: if we add 0 with 1 the result will still be 1.

0 + 1 = 1

Whereas, when we add the opposite value of the same number is said to be an inverse element. The result of the addition of inverse elements is an identity element that is 0. For example, if we add 1 with its opposite value -1, then the result would be

1 + (-1) = 0

Subtraction(-)

Subtraction is the inverse of addition. It is the arithmetic operation that determines the difference between two values (i.e. minuend minus the subtrahend) is subtraction. In the cases, where the minuend is greater than the subtrahend, the difference is positive.

5 – 2 = 3

While, if the subtrahend is greater than minuend the difference between them will be negative.

2 – 5 = -3

Multiplication (×)

The operation which combines two values to give a single product of them is called multiplication. The two values involved in the operation of multiplication are known as multiplicand and multiplier.

The product of two values is supposedly p and q is expressed in p.q or p×q form.

Division (÷)

The division is the inverse of multiplication. It is the operation that computes the quotient of two values. The two values involved in it are known as dividends by the divisor. If the quotient is more than 1, if the dividend is greater than the divisor the result would be a positive number.

9/3 = 3

Now, let’s jump onto the question,

What two numbers have a sum of 7 and a difference of 1?

Answer:

Let the two numbers be x and y.

According to the question,

x + y = 7……(i)

x – y = 1…….(ii)

Now, to know the value of one of the variables we need to solve one of the equations given above and substitute the value in the other equation.

Now, from equation (i) we can obtain the value of x by isolating it.

=>x = 7 – y

Then, substituting the value of x in equation (ii)

=>7 – y – y = 1

=>7 – 2y = 1

=>2y = 7 – 1

=>y = 6/2 = 3

Substituting the value of y = 3 in x = 7 – y

=>x = 7 – 3 = 4

Hence, the two variables are x = 4 and y = 3.

Question 1: The sum of two numbers is 8 and their difference is 2. Find the variables.

Solution:

Let the two numbers be x and y.

According to the question,

x + y = 8……(i)

x – y = 2…….(ii)

Now, to know the value of one of the variables we need to solve one of the equations given above and substitute the value in the other equation.

Now, from equation (i) we can obtain the value of x by isolating it.

=>x = 8 – y

Then, substituting the value of x in equation (ii)

=>8 – y – y = 2

=>8 – 2y = 2

=>2y = 8 – 2

=>y = 6/2 = 3

Substituting the value of y = 3 in x = 8 – y

=>x = 8 – 3 = 5

Hence, the two variables are x = 5 and y = 3.

Question 2: Let’s take two numbers whose sum is 11 and their product is 30. Find the value of numbers.

Solution:

Let the two numbers be x and y.

Now, according to the question,

x + y = 11……(i)

x.y = 30…….(ii)

Now, from equation (ii) we can obtain the value of x by isolating it.

=>x = 30/y

Then, substituting the value of x in equation (i)

=>30/y + y = 11

=>30 + y2 = 11y

=>y2 – 11y + 30 = 0

=>y2 – 6y – 5y + 30 = 0

=>y(y – 6) – 5(y – 6) = 0

=>(y – 6)(y – 5) = 0

=>y = (5, 6)

=>x = (6, 5)

Question 3: The sum of the two numbers is 8 and the sum of their reciprocal is 8/15. Find the numbers.

Solution:

Let the two numbers be x and y.

According to the question,

x + y = 8…….(i)

1/x + 1/y = 8/15…….(ii)

Now, to know the value of one of the variables we need to solve one of the equations given above and substitute the value in the other equation.

Now, from equation,

=>8/xy = 8/15

=>xy = 15

=>x = 15/y

Then, substituting the value of x in equation (i)

=>15/y + y = 8

=>15 + y2 = 8y

=>y2 – 8y + 15 = 0

=>(y – 5)(y – 3) = 0

=>y = (5, 3)

=>x = (3, 5)

Question 4: If the sum of two non-zero numbers is 8 and their difference is 7. Find the value of numbers.

Solution:

Let the two numbers be x and y.

According to the question,

x + y = 25……(i)

x – y = 7…..(ii)

Now, to know the value of one of the variables we need to solve one of the equations given above and substitute the value in the other equation.

Now, from equation (i) we can obtain the value of x by isolating it.

=>x = 25 – y

Then, substituting the value of x in equation (ii)

=>25 – y – y = 7

=>25 – 2y = 7

=>2y = 25 – 7

=>2y = 18

=>y = 18/2 = 9

Substituting the value of y = 9 in x = 25 – y

=>x = 25 – 9 = 16

Hence, the two variables are x = 16 and y = 9.

Mark, a math enthusiast, loves writing tutorials for stumped students and those who need to brush up on their math skills.

Read on for an easy explanation of these concepts, along with helpful examples.

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If you are asked to work out the product of two or more numbers, then you need to multiply the numbers together. If you are asked to find the sum of two or more numbers, then you need to add the numbers together.

Below, we will work through several examples together.

In a Nutshell

"Product" means multiply.

"Sum" means add.

What are the product and sum of 6 and 8?

• The product of 6 and 8 is 6 × 8 = 48
• The sum of 6 and 8 is 6 + 8 = 14

What are the product and sum of 9 and 3?

Work out the product and sum of 9 and 3.

• The product of 9 and 3 is 9 × 3 = 27
• The sum of 9 and 3 is 9 + 3 = 12

What are the product and sum of -7 and 9?

• The product of -7 and 9 is -7 × 9 = -63
• The sum of -7 and 9 is -7 + 9 = 2

Write down two numbers that have a product of 42.

All you need to do is look for two numbers that multiply to give 42. There are several possible answers:

• 1 and 42
• 2 and 21
• 3 and 14
• 6 and 7
• -1 and -42
• -2 and -21
• -3 and -14
• -6 and -7

What are the product and sum of 2, 4 and 9?

Work out the product of 2, 4 and 9.

The product means that you need to multiply the three numbers together.

2 × 4 × 9 = 72

The sum means that you need to add the three numbers together.

2 + 4 + 9 = 15

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

Questions & Answers

Question: What do you get if you double 15?

Answer: Just multiply the number by 2 to give 30.

Question: How do you multiply the sum of 12 and 8 by their difference?

Answer: The sum of 12 and 8 is 12 add 8 which is 20.

The difference between 12 and 8 is 12 minus 8 which is 4.

Now 20 times 4 is 80.

Question: what is the sum and product of -5 and -12?

Answer: Sum means you need to add the numbers (-5 + -12 = -5 - 12 = -17).

Product means you need to multiply the numbers (-5 multiplied by -12 = 60).

Question: Eight times the sum of 5 and some number is 184. What is the number?

Answer: First divide 184 by 8 to give 23.

Next subtract 5 from 23 to give 18.

So the answer is 18.

Question: What is the product of 18, 60 and 78?

Answer: To work out the product just multiply all of the 3 numbers to give 84240.

Question: What is 7/4 x 12?

Answer: Multiply can mean "of" so work out 7/4 of 12.

Do this by dividing 12 by 4 to give 3, and multiplying 3 by 7 to give 21.

Question: What two numbers have the product of 10, and sum of 11?

Answer: The two numbers are 1 and 10, because 1 multiplied by 10 is 10, and 1 added to 10 is 11.

Question: What is the product of 6062 and 505?

Answer: To find the product multiply 6062 by 505 to give 3,061,310.

Question: Josh added 134.086 to 561.27. What digit was in the hundredths place in the sum?

Answer: The answer to the sum of these numbers is 695.356.

The hundredths column is the second number after the decimal point which is 5.

Question: Tom thinks of a number between 15 and 30. The sum of the digits is 7. What is toms number?

Answer: The number could be 16 or 25, since 1 add 6 is 7, and 2 add 5 is also 7.

Question: What would the product be for 669 and 83?

Answer: Just multiply 669 by 83 to give 55527

Question: If the quotient of - 56 and 8 is subtracted from the product of - 12 and - 3, what is the difference?

Answer: Divide -56 by 8 to give -7.

Multiply -12 and -3 to give 36.

Now the difference between -7 and 36 is 43.

Question: What are the two numbers that have a product of 132 and sum of -23?

Answer: The two numbers are -12 and -11, because -12 times -11 is 132, and -12 + -11 = -23.

Question: The sum of two numbers is 50. One number is 6 more than the other. What are the numbers?

Answer: Half of 50 is 25, and half of 6 is 3.

Now 25 - 3 is 22, and 25 + 3 is 28.

So the two numbers are 22 and 28 as they both add up to 50 and 28 is 6 more than 22.

Question: What is the sum of k and 4?

Answer: k + 4, as sum means add.

Question: Can you find the product of 2/9 of 90?

Answer: Just divide 90 by 9 to give 10, and multiply 10 by 2 to give 20.

However, if you have a scientific calculator type in 2/9 times 90 using your fraction button.

Question: What is The product of 48 × 9?

Answer: Product means multiply, so with a calculator type in 48 times 9.

If doing mentally, first do 40 multiplied by 9 which is 360, and then do 8 multiplied by 9 which is 72. So 360 + 72 = 432.

Question: What is 5 added to the product of 4 and 6?

Answer: First follow BODMAS and work out the product of 4 and 6 by multiplying these numbers together to give 24.

Next add the 4 onto 24 to give 28.

Question: Find the product of five and ten and divide it by five. What's the answer?

Answer: First work out the product of 5 and 10 to give 50 (multiply 5 by 10).

Now divide the answer by 5 to give 10.

Question: What two numbers have a product of 60 and a sum of -19?

Answer: The two numbers are -4 and -15.

This is because -4 added to -15 gives -19, and -4 times -15 gives 60.

Question: What are two numbers that have a product of 3/8?

Answer: You need to figure out two fractions which multiply to give 3/8.

The two fractions are 3/4 and 1/2.

Question: Two numbers have a sum of 5 and a product of 6 what are the numbers?

Answer: The two numbers are 3 and 2.

This is becuase 3 + 2 gives 5, and 3 multiplied by 2 gives 6.

Question: What is the product of 1 3/ 5 and 8 1 /2 ?

Answer: Change these mixed numbers to decimals as it will be easier to work out (1 3/5 = 1.6 and 8 1/2 = 8.5). So the product is 1.6 multiplied by 8.5 which is 13.6. The sum will be 1.6 + 8.5 = 10.1.

Question: Can you find two numbers which have a sum of 13 and a product of 12?

Answer: Sum means add and product means multiply.

The two numbers are 1 and 12, as 1 + 12 = 13 and 1 multiplied by 12 gives 12.

Question: What are the product, sum, and difference between 123 and 89?

Answer: The sum of 123 and 89 is 123 + 89 which is 212.

The difference os 123 and 89 is 123 - 89 which is 34.

Now the product of 212 and 34 is 212 times 34 which is 7208.

Question: What is 1/3 the sum of six and three?

Answer: First of all, work out the sum of 6 and 3 by adding these together to give 9.

Now to find a third of something you divide by 3, so 9 divided by 3 is 3.

Question: Can you find two numbers which have a product of 12 and a sum of 7?

Answer: 4 and 3 are a the answers.

4 multiplied by 3 is 12. 4 add 3 is 7.

Question: How can I work out the product of 18 and 4?

Answer: All you need to do is 18 multiplied by 4 which is 72.

Question: Can you find two numbers which have a product of 8 and a sum of 7?

Answer: Yes, you can but the answers are not integers (whole numbers).

You will need to solve the simultaneous equations xy = 8 and x + y = 7.

Question: The product of 5 whole numbers is 3. What is the sum of these numbers?

Answer: Product means multiply, so the 5 whole numbers that multiply together to give 3 are 1,1,1,1 and 3 (that is fours 1's and one 3).

Since "sum" means "add," then 1 + 1 + 1 + 1 + 3 gives a final answer of 7.

Question: Can you find two numbers which have a product of 741 and sum of 70 ?

Answer: The two number you are looking for are 57 and 13.

This is beacause 57 multiplied by 13 gives 741, and 57 added to 13 gives 70.

Question: What are the two numbers that the sum is 12 and the product is 36?

Answer: Remember, sum means add and product means multiply.

The two numbers that answer this question are 6 and 6.

This is because 6 add 6 is 12, and 6 multiplied by 6 is 36.

Question: The product of 4/3 and a number is 84. What is the number?

Answer: Divide 84 by 4/3 to give 63.

Question: What number pair has the sum of 5 product of 4?

Answer: Sum means add, and product means multiply.

Therefore the two numbers needed are 1 and 4, because 1 added to 4 is 5 and 1 multiplied by 4 is 4.

Question: What are the two numbers which have a sum and product of 1 and 9?

Answer: The sum can be found by adding 1 and 9 to give 10.

The product of 1 and 9 can be found by multiplying 1 and 9 to give 9.

Question: Can you find two numbers that have a product of -18 and a sum of -7?

Answer: The two numbers are -9 and 2.

This is because -9 multiplied by 2 is -18, and -9 plus 2 is -7.

Question: What is the sum of 2 and 7?

Answer: The two numbers needed to be added together if you are working out the sum.

So 2 + 7 = 9.

Question: Can you find a number which has a product of -14 and a sum of 5?

Answer: The two numbers are -2 and 7, since -2 times 7 is -14, and -2 + 7 is 5.

Question: The product of two numbers is 72 and the sum is 18, what are that two numbers?

Answer: The two numbers are 6 and 12, because 6 times 12 is 72, and 6 added to 12 is 18.

Question: What is the answer to halve the product of 5 and 12?

Answer: First of all work out the product of 5 and 12.

Therefore, multiply the 5 by the 12 to give 60 (since product means multiply).

Now half 60 to give a final of answer of 30.

Question: What is the product of 0.009 & 2.84?

Answer: To find the product of any two numbers just multiply them.

0.009 multipled by 2.84 gives 0.02556.

Question: How can I work out the sum of 2/12 and 6/12?

Answer: To find the sum just add the two fractions together to get 8/12 which simplifies to 2/3.

Question: Can you find two numbers which have a product of 60 and a sum of 100?

Answer: The two numbers are not integers, so there is not an obvious answer.

You will need to solve the simultaneous equations xy = 60, and x + y = 100, to give x

The two answers are 99.3963 and 0.6036 rounded to 4 decimal places.

Question: The product of two whole numbers is 648 and their sum is 51. What are the two numbers?

Answer: The two numbers are 24 and 27.

This is because 24 multiplied by 27 gives 648.

Also, 24 added to 27 gives 51.

Remember, product means multiply and sum means add.

Question: What is the sum of 19 and 25 multiplied by their difference?

Answer: First, work out the sum of 19 and 25 by adding the two numbers together to give 44.

Next, find the difference of 25 and 19 by subtracting the numbers to give 6.

Now multiply 44 by 6 to give 264.

Question: What’s the product of 7/12 and 14?

Answer: Convert 7/12 into a decimal by dividing 7 by 12 to give 0.583..., now multiply this by 14 to give 8.17 rounded to 2 decimal places or 49/6 if you prefer a fraction.

Question: What is the product of 1 and 5?

Answer: Just multiply 1 by 5 to give an answer of 5.

Question: What two numbers have a sum of 15 and a product of 36 ?

Answer: The answers are 3 and 12, since 3 + 12 = 15, and 3 multiplied by 12 is 36.

Question: Can you find the two numbers which have the product of 12 and a sum of -7?

Answer: The two numbers you are looking for are -4 and -3.

This is because -4 times -3 is 12, and -4 + -3 is -7.

Question: The sum of two numbers is 9 and their difference is 3. What is their product?

Answer: First find out what the two numbers are.

These are 6 and 3, since 6 add 3 is 9, and 6 minus 3 is 3.

Now multiply 6 by 3 to give 18.

© 2011 Mark

Rachel Simpson on April 30, 2020:

What is the product of 2,3and 4