1. The sum of a number and 2 is 6 less than twice that number.
2. A rectangular garden has a width that is 8 feet less than twice the length. Find the dimensions if the perimeter is 20 feet.
4. Six times a number is less than 72. What numbers satisfy this condition?
5. A number increased by 10 is greater than 50. What numbers satisfy this condition?
I am confused as to how to convert them into equations. These are not test questions I promise. Can someone please help clear the darkness a little?
In many of these it is important to see what "verbiage" can be ignored and to key in on the critical phrases.
e.g. #2
- "rectangular" is important since we need the perimeter
- "a width that is 8 feet less than twice the length"
The reference was "the length", so let the length be x
I see "twice the length" ----> 2x
I see "8 feet less than 2x -----> 2x - 8
I know the perimeter is : twice the length + twice the width
= 2(x) + 2(2x-8)
But we are told that equals 20 , so
2x + 2(2x-8) = 20
Now just solve that
let's try another:
#5 A number increased by 10 is greater than 50.
" a number" ---- x
"increased by 10" ---- +10
"is greater than" ----- >
x + 10 > 50
etc
Simplification
1.
x + 2 = 2 x - 6,
2.
W = 2 L - 8
P = 2 ( L + W ) = 2 ( L + 2 L - 8 ) = 2 ( 3 L - 8 ) = 6 L - 16 = 20
6 L -16 = 20
4.
6 x < 72
5.
x + 10 > 50
Try to solve this equations.
Looking at #1,
-- The sum of a number and 2 is 6 less than twice that number --
Some students find this type of problem confusing. Do I add or subtract that 6?
I know there is a < somewhere.
I suggested that they rephrase the problem:
"The sum of a number and 2 is less than twice that number by 6 "
x + 2 < 2x by 6
so right now the 2x is the bigger part by 6
so if we subtract 6 from the right side, they would be equal, so
x + 2 = 2x - 6
x = 8
check:
the sum of the number and 2 : 8 + 2 = 10
twice the number: 16
is 10 less than 16 by 6 ? YES
I am a 3 - digit number. I have 6 in the ones place. I am greater than 127 but less than 142. What am I?
Sure, I'd be happy to help clear the darkness a little! Let's convert these statements into equations.
1. The sum of a number and 2 is 6 less than twice that number.
Let's call the number "x". The sum of the number and 2 can be written as "x + 2". We know that this is 6 less than twice the number, so we can express that as "2x - 6". Therefore, our equation is:
x + 2 = 2x - 6
2. A rectangular garden has a width that is 8 feet less than twice the length. Find the dimensions if the perimeter is 20 feet.
Let's call the length of the garden "L" and the width "W". The width is 8 feet less than twice the length, so we can write that as "W = 2L - 8". The perimeter of a rectangle is given by P = 2L + 2W. We know that the perimeter is 20 feet, so we can write the equation as:
2L + 2W = 20
3. Six times a number is less than 72. What numbers satisfy this condition?
Let's call the number "x". We are told that six times the number is less than 72, so we can write that as "6x < 72". This inequality tells us that the number needs to be less than 12 for it to satisfy the condition.
4. A number increased by 10 is greater than 50. What numbers satisfy this condition?
Again, let's call the number "x". We are told that the number increased by 10 is greater than 50, so we can write that as "x + 10 > 50". This inequality tells us that the number needs to be greater than 40 for it to satisfy the condition.
I hope these explanations help! Let me know if you have any more questions or need further clarification.
Of course! Let's break down each problem and convert them into equations.
1. The sum of a number and 2 is 6 less than twice that number.
Let's start by assigning a variable to the unknown number.
Let's say the number is x.
The sum of the number (x) and 2 can be written as (x + 2).
Twice that number is 2 times x, which is 2x.
The problem states that the sum of the number and 2 is 6 less than twice that number. So we can write it as:
(x + 2) = 2x - 6
2. A rectangular garden has a width that is 8 feet less than twice the length. Find the dimensions if the perimeter is 20 feet.
Let's represent the length of the garden with the variable L.
The problem states that the width (W) is 8 feet less than twice the length, which can be written as (2L - 8).
To find the perimeter of a rectangle, we add up all four sides. In this case, we have two lengths (L + L) and two widths (W + W).
The perimeter is given as 20 feet, so we can write it as:
2L + 2W = 20
Substitute the value of W from the previous equation:
2L + 2(2L - 8) = 20
4. Six times a number is less than 72. What numbers satisfy this condition?
If we let the unknown number be x, we can express "six times a number" as 6x.
The problem states that this value (6x) is less than 72, so we can write it as:
6x < 72
5. A number increased by 10 is greater than 50. What numbers satisfy this condition?
Let's assign the unknown number as x.
The problem states that this number increased by 10 is greater than 50, so we can write it as:
x + 10 > 50
These equations should help you solve the problems! Substitute the variables with the appropriate values, and solve for the unknown variables to find the answers to the questions.