1.9.3 Bounds
To describe all the possible values that a rounded number could be, we use bounds.
Upper and lower bounds:
The lower bound is the smallest value that would round up to the estimated value.
While the upper bound is the smallest value that would round up to the next estimated value.
The bounds for the number, \( x \), can be written as
\( Lower \ bound \leq x < Upper \ bound \)
Keep in mind that the lower bound is included in the range of possible values for \( x \), while the upper bound is not.
1.9.3.1 Working with bounds
To calculate upper and lower bound of a number,\( x \).
For upper bound add on half the degree of accuracy
For lower bound take off half the degree of accuracy
Write the number as
\( Lower \ bound \leq x < Upper \ bound \)
Bounds of a calculation
Addition:
If you are adding two numbers together: \( X=a+b \)
The upper bound of \( X \) can be obtained by adding the upper bounds of \( a \) and \( b \) together.
The lower bound of \( X \) can be obtained by adding the lower bounds of \( a \) and \( b \) together.
Subtraction:
If you are subtracting a number from another number: \( X=a-b \)
The upper bound of \( X \) can be found by subtracting the lower bound of \( b \) from upper bound of \( a \)
The lower bound of \( X \) can be obtained by subtracting the upper bound of \( b \) from lower bound of \( a \)
Multiplication:
If you are multiplying two numbers together: \( X=a \times b \)
The upper bound of \( X \) can be obtained by multiplying upper bounds of \( a \) and \( b \) together
The lower bound of \( X \) can be obtained by multiplying lower bound of \( a \) and \( b \) together
Division:
If you are dividing a number by another number: \( X=a \div b \)
The upper bound of \( X \) can be obtained by dividing the upper bound of \( a \) by lower bound of \( b \)
The lower bound of \( X \) can be obtained by dividing the lower bound of \( a \) by upper bound of \( b \)
Worked example:
Priya has \( 50 \) identical parcels.
Each parcel has a mass of \( 17 \) kg, correct to the nearest kilogram.
Find the upper bound for the total mass of \( 50 \) parcels.
Degree of accuracy is \( 1 \)
Half the degree of accuracy
\( 1 \div 2=0.5 \)
For upper bound, add on half the degree of accuracy
\( 17+0.5=17.5 \)
Upper bound for the total mass of \( 50 \) parcels
\( 50 \times 17.5=875 \)
\( 875 \)
Test yourself
Question 1:
Calculate an estimate for \( \frac{17.3 \times 3.81 }{11.5} \).
State, with a reason, whether the estimate is an overestimate or an underestimate.
[2]
Question 2:
The sides of a regular hexagon are \( 80 \) mm, correct to the nearest millimeter.
Calculate the lower bound of the perimeter of the hexagon.
[2]
Question 3:
\( P=2w+h \)
\( w=12 \) correct to the nearest whole number.
\( h=4 \) correct to the nearest whole number.
Work out the upper bound for the value of \( P \).
[2]
Question 4:
The length of the side of a square is \( 12 \) cm, Correct to the nearest centimeter.
Calculate the upper bound for the perimeter of the square.
[2]
Jo measures the length of a rope and records her measurement correct to the nearest ten centimeters.
The upper bound for er measurement is \( 12.53 \)m.
Write down the measurement she record.
[1]