1.7.7

Upper Lower Bounds

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Upper and Lower Bounds

When we estimate answers, we will either overestimate or underestimate the correct answers. These estimations can give us a range which the correct answer must be within.

What is 5.43 & times 12?

What is 5.43 & times 12?

  • This multiplication is hard to work out, but we can find a range for the answer by rounding 5.43.
Upper bound

Upper bound

  • If we round 5.43 up to the nearest whole number, we will overestimate the answer.
    • We know that the real answer is less than this overestimate so we can write (5.43 × 12) < 72.
Lower bound

Lower bound

  • If we round 5.43 down to the nearest whole number we will underestimate the answer.
    • We know that the real answer is more than this overestimate so we can write 60 < (5.43 × 12).
Inequality

Inequality

  • From these estimations we know that 5.43 &times 12 is more than 60 and less than 72.
    • We can write this using the double inequality: 60 < (5.43 × 12) < 72.
Jump to other topics
1

Number

1.1

Place Value

1.1.1Ordering Numbers1.1.2Ordering Numbers - Practice1.1.3Comparing Numbers1.1.4Comparing Numbers - Practice1.1.5Equals Symbols1.1.6Equals Symbols - Practice1.1.7Adding Negative Numbers1.1.8Adding Negative Numbers - Practice1.1.9Subtracting Negative Numbers1.1.10Multiplying Negative Numbers1.1.11Dividing Negative Numbers1.1.12End of Topic Test - Place Value
1.2

Factors & Multiples

1.2.1Factors1.2.2Factors - Practice1.2.3Prime Numbers1.2.4Prime Numbers - Practice1.2.5Multiples1.2.6Multiples - Practice1.2.7Prime Factors1.2.8Prime Factors - Practice1.2.9Highest Common Factors1.2.10Highest Common Factors - Practice1.2.11Highest Common Factor Prime Factors1.2.12Highest Common Factor Prime Factors - Practice1.2.13Lowest Common Multiples1.2.14Lowest Common Multiples - Practice1.2.15Lowest Common Multiples Prime Factors1.2.16Lowest Common Multiples Prime Factors - Practice1.2.17End of Topic Test - Factors & Multiples
1.3

Operations

1.3.1Addition1.3.2Addition - Practice1.3.3Subtraction1.3.4Subtraction - Practice1.3.5Multiplication 10,100...1.3.6Multiplication 10,100... - Practice1.3.7Grid Multiplication1.3.8Grid Multiplication - Practice1.3.9Division by 10s1.3.10Division by 10s - Practice1.3.11Powers1.3.12Powers - Practice1.3.13Roots1.3.14Roots - Practice1.3.15Surds1.3.16Surds - Practice1.3.17Rationalise denominator1.3.18Rationalise Denominator - Practice1.3.19Simplify Surds1.3.20Simplify Surds - Practice1.3.21BIDMAS1.3.22BIDMAS - Practice1.3.23Inverse Ops1.3.24Inverse Ops - Practice1.3.25Standard Form1.3.26Addition in Standard Form1.3.27Subtraction in Standard Form1.3.28End of Topic Test - Operations1.3.29End of Topic Test - Standard Form
1.4

Fractions, Decimals & Percentages

1.4.1Fractions1.4.2Fractions - Practice1.4.3Mixed & Improper Fractions1.4.4Mixed & Improper Fractions - Practice1.4.5Write 2 Numbers as Fractions1.4.6Write 2 Numbers as Fractions - Practice1.4.7Adding Fractions1.4.8Adding Fractions - Practice1.4.9Subtracting Fractions1.4.10Multiplying Fractions1.4.11Multiplying Fractions - Practice1.4.12Dividing Fractions1.4.13Finding Fractions1.4.14Finding Fractions - Practice1.4.15Improper & Mixed Fractions1.4.16Decimal Place Value1.4.17Decimal Place Value - Practice1.4.18Fraction to Decimal1.4.19Fraction to Decimal - Practice1.4.20Find Percentage Unitary1.4.21Find Percentage Unitary - Practice1.4.22Fraction to Percentage1.4.23Fraction to Percentage - Practice1.4.24Percentage to Fraction1.4.25Percentage to Decimal1.4.26Percentage to Decimal - Practice1.4.27Decimal to Percentage1.4.28Decimal to Percentage - Practice1.4.29Percentage using Decimal1.4.30End of Topic Test - Fractions1.4.31End of Topic Test - Decimals
1.5

Growth & Decay

1.5.1Population Growth1.5.2Population Growth - Practice1.5.3Compound Growth & Decay1.5.4Compound Growth & Decay - Practice1.5.5Compound Interest1.5.6Compound Interest - Practice1.5.7Compound Interest 21.5.8Compound Interest 2 - Practice1.5.9Depreciation1.5.10Depreciation - Practice1.5.11End of Topic Test - Growth & Decay
1.6

Measurements & Units

1.6.1Standard Units 11.6.2Standard Units 1 - Practice1.6.3Standard Units 21.6.4Standard Units 2 - Practice1.6.5Standard Units 31.6.6Standard Units 3 - Practice1.6.7Length1.6.8Area1.6.9Volume1.6.10Capacity1.6.11Time1.6.12Time - Practice1.6.13Money1.6.14Mass1.6.15End of Topic Test - Measurements & Units
1.7

Accuracy

1.7.1Decimal Rounding1.7.2Decimal Rounding - Practice1.7.3SigFig Rounding1.7.4SigFig Rounding - Practice1.7.5Estimations1.7.6Estimations - Practice1.7.7Upper Lower Bounds1.7.8Upper Lower Bounds - Practice1.7.9End of Topic Test - Accuracy
2

Algebra

2.1

Introduction to Algebra

2.1.1Algebra Basics2.1.2Algebra Basics - Practice2.1.3Algebraic Concepts2.1.4Algebraic Concepts - Practice2.1.5Writing Expressions2.1.6Writing Expressions - Practice2.1.7Collecting Terms2.1.8Collecting Terms - Practice2.1.9Common Factors2.1.10Common Factors - Practice2.1.11Expanding Brackets2.1.12Expanding Brackets - Practice2.1.13FOIL2.1.14FOIL - Practice2.1.15Rearranging Formulae2.1.16Rearranging Formulae - Practice2.1.17End of Topic Test - Introduction to Algebra
2.2

Graphs

2.2.1Coordinates2.2.2Coordinates - Practice2.2.3Gradient and Intercept2.2.4Gradient and Intercept - Practice2.2.5Calculate Gradient &Y-intercept2.2.6Calculate Gradient & Y-intercept2.2.7Finding Equation2.2.8Finding Equation - Practice2.2.9Quadratic Graphs2.2.10Quadratic Graphs - Practice2.2.11End of Topic Test - Graphs
2.3

Solving Equations

2.3.1Solve Linear Equations2.3.2Solve Linear Equations - Practice2.3.3Words To Equation2.3.4Words To Equation - Practice2.3.5Solve Linear Equations Graphically2.3.6Solve Linear Equations Graphically - Practice2.3.7Solve Inequality2.3.8Solve Inequality - Practice2.3.9Graphing Inequalities2.3.10Graphing Inequalities - Practice2.3.11Simultaneous Equations2.3.12Simultaneous Equations - Practice2.3.13Simultaneous Equations Graph2.3.14Simultaneous Equations Graph - Practice2.3.15Quadratic +/-2.3.16Quadratic +/- - Practice2.3.17Factorise Quadratics2.3.18Factorise Quadratics - Practice2.3.19Complete the Square2.3.20Complete the Square - Practice2.3.21End of Topic Test - Graphs & Equations2.3.22End of Topic Test - Solving Equations
2.4

Sequences

2.4.1Arith & Geo Sequences2.4.2Arith & Geo Sequences - Practice2.4.3Key Sequences2.4.4Key Sequences - Practice2.4.5Nth Term Rules2.4.6Nth Term Rules - Practice2.4.7Find NthTerm in Linear Sequence2.4.8Find NthTerm in Linear Sequence - Practice2.4.9Find Nth Term in Quadratic Sequence2.4.10Find Nth Term in Quadratic Sequence - Practice2.4.11End of Topic Test - Sequences
3

Ratio, Proportion & Rates of Change

3.1

Manipulating Ratios

3.1.1Manipulating Ratios3.1.2Manipulating Ratios - Practice3.1.3Turn Ratios into Equations3.1.4Turn Ratios into Equations - Practice3.1.5Part:Part Ratio3.1.6Part:Part Ratio - Practice3.1.7Part:Whole Ratio3.1.8Part:Whole Ratio - Practice3.1.9Differences in Ratios3.1.10Differences in Ratios - Practice3.1.11Fractions in Ratios3.1.12Fractions in Ratios - Practice3.1.13Ratios to Fractions3.1.14Ratios to Fractions - Practice3.1.15End of Topic Test - Manipulating Ratios
3.2

Ratios in Practice

3.2.1Covert Units3.2.2Covert Units - Practice3.2.3Formula Triangles3.2.4Formula Triangles - Practice3.2.5Scale Factors3.2.6Scale Factors - Practice3.2.7Ratios in Shapes3.2.8Ratios in Shapes - Practice3.2.9End of Topic Test - Ratios
3.3

Percentage & Finance

3.3.1Percentage Increase/Decrease3.3.2Percentage Increase/Decrease - Practice3.3.3Percentage Change3.3.4Percentage Change - Practice3.3.5Original Value3.3.6Original Value - Practice3.3.7Profit, Interest & Discount3.3.8Profit, Interest & Discount - Practice3.3.9Simple Interest3.3.10Simple Interest - Practice3.3.11End of Topic Test - Percentage & Finance
3.4

Proportion

3.4.1Proportion3.4.2Proportion - Practice3.4.3Direct Proportion3.4.4Direct Proportion - Practice3.4.5Inverse Proportion3.4.6Inverse Proportion - Practice3.4.7End of Topic Test - Percentages & Proportion
4

Geometry & Measures

4.1

Introduction to Geometry

4.1.1LinesAngles14.1.2LinesAngles1 - Practice4.1.3LinesAngles24.1.4LinesAngles2 - Practice4.1.5Regular Polygons4.1.6Regular Polygons - Practice4.1.7Triangles and Angles4.1.8Triangles and Angles - Practice4.1.9Quadrilaterals4.1.10Quadrilaterals - Practice4.1.11Midpoints4.1.12Midpoints - Practice4.1.13Bisectors and Perpendiculars4.1.14Bisectors and Perpendiculars - Practice4.1.15Angle Rules4.1.16Angle Rules - Practice4.1.17Angles in Parallel Lines4.1.18Angles in Parallel Lines - Practice4.1.19Angles in a Polygon4.1.20Angles in a Polygon - Practice4.1.21Angles in a Triangle4.1.22Angles in a Triangle - Practice4.1.23Bearings4.1.24Bearings - Practice4.1.25End of Topic Test - Geometry4.1.26End of Topic Test - Geometry & Measures
4.2

Ruler & Compass Constructions

4.2.1Midpoint4.2.2Bisectors and Perpendiculars4.2.3Loci4.2.4End of Topic Test - Ruler & Compass Constructions
4.3

Perimeter & Area

4.3.1Perimeter4.3.2Perimeter - Practice4.3.3Area4.3.4Area - Practice4.3.5End of Topic Test - Perimeter & Area
4.4

Triangles, Quadrilaterals & Circles

4.4.1Types of Triangle4.4.2Types of Triangle - Practice4.4.3Angles & Triangles4.4.4Angles & Triangles - Practice4.4.5Congruence4.4.6Congruence - Practice4.4.7Congruent Triangles4.4.8Congruent Triangles - Practice4.4.9Similar Triangles4.4.10Similar Triangles - Practice4.4.11Scale Factors & Enlargement4.4.12Scale Factors & Enlargement - Practice4.4.13Properties of Circle 14.4.14Properties of Circle 1 - Practice4.4.15Properties of a Circle 24.4.16Properties of a Circle 2 - Practice4.4.17Area of a Circle4.4.18Area of a Circle - Practice4.4.19Circumference of a Circle4.4.20End of Topic Test - Triangles, Quadrilaterals4.4.21End of Topic Test - Circles
4.5

Transformations

4.5.1Types of Transformations4.5.2Types of Transformations - Practice4.5.3Effects of Transformations4.5.4Effects of Transformations - Practice4.5.5End of Topic Test - Transformations
4.6

Triangle Formulae

4.6.1Pythagoras' Theorem4.6.2Pythagoras' Theorem - Practice4.6.3Trigonometric Ratios4.6.4Trigonometric Ratios - Practice4.6.5End of Topic Test - Triangle Formulae
4.7

3D Shapes

4.7.1Types of 3D Shape4.7.2Types of 3D Shape - Practice4.7.3Plans & Elevations4.7.4Plans & Elevations - Practice4.7.5Nets4.7.6Nets - Practice4.7.7Volume of 3D Shapes4.7.8Volume of 3D Shapes - Practice4.7.9Volume of a Cuboid4.7.10Volume of a Cylinder4.7.11Volume of a Sphere4.7.12Volume of a Hemisphere4.7.13Volume of a Cone4.7.14Volume of a Pyramid4.7.15Surface Area of a Cuboid4.7.16Surface Area of a Cylinder4.7.17Surface Area of a Sphere/Hemisphere4.7.18Surface Area of a Cone4.7.19Surface Area of Composite Shapes4.7.20Surface Area of Composite Shapes - Practice4.7.21End of Topic Test - Triangles, Area, & 3D Shapes4.7.22End of Topic Test - 3D Shapes
4.8

Vectors

4.8.1Vector Introduction4.8.2Vector Introduction - Practice4.8.3Vectors4.8.4Vectors - Practice4.8.5End of Topic Test - Vectors
5

Probability

5.1

Introduction to Probability

5.1.1Probability5.1.2Probability - Practice5.1.3Probability scale5.1.4Probability Scale - Practice5.1.5Randomness & Fairness5.1.6Randomness & Fairness - Practice5.1.7Sum of Probabilities5.1.8Sum of Probabilities - Practice5.1.9Relative & Expected Frequency5.1.10Relative & Expected Frequency - Practice5.1.11Frequency Tables5.1.12Frequency Tables - Practice5.1.13Use Frequency Trees5.1.14Use Frequency Trees - Practice5.1.15Complete Frequency Trees5.1.16Complete Frequency Trees - Practice5.1.17End of Topic Test - Probability
5.2

Representing Probability

5.2.1Sample Space Diagrams5.2.2Sample Space Diagrams - Practice5.2.3Sets5.2.4Sets - Practice5.2.5Venn Diagrams5.2.6Venn Diagrams - Practice5.2.7End of Topic Test - Representing Probability
5.3

Probability Calculations

5.3.1Theoretical Probability5.3.2Theoretical Probability - Practice5.3.3Venn Diagram Probabilities5.3.4Venn Diagram Probabilities - Practice5.3.5Multiple Events5.3.6Multiple Events - Practice5.3.7Dependent & Independent Events5.3.8Dependent & Independent Events - Practice5.3.9Calculating Probabilities5.3.10Calculating Probabilities - Practice5.3.11Complex Tree Diagrams5.3.12Complex Tree Diagrams - Practice5.3.13End of Topic Test - Probability
6

Statistics

6.1

Introduction to Statistics

6.1.1Statistics Basics6.1.2Statistics Basics - Practice6.1.3Statistics Basics 26.1.4Statistics Basics 2 - Practice6.1.5Comparing Data6.1.6Comparing Data - Practice6.1.7Comparing Datasets6.1.8Comparing Datasets - Practice6.1.9Writing Comparisons6.1.10Writing Comparisons - Practice6.1.11Frequency Table6.1.12Frequency Table - Practice6.1.13End of Topic Test - Statistics
6.2

Charts & Graphs

6.2.1Bar Charts6.2.2Bar Charts - Practice6.2.3Pie Charts6.2.4Pie Charts - Practice6.2.5Pictograms6.2.6Pictograms - Practice6.2.7Scatter Graphs6.2.8Scatter Graphs - Practice6.2.9Making Predictions6.2.10Making Predictions - Practice6.2.11Correlation and Causation6.2.12Correlation and Causation - Practice6.2.13Outliers6.2.14Outliers - Practice6.2.15End of Topic Test - Statistics

Practice questions on Upper Lower Bounds

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

  1. 1
    By rounding to 2 significant figures, find the upper and lower bounds for 12.53 &times; 3.Multiple choice
  2. 2
    By rounding to 1 significant figure, find the upper and lower bounds for 9.67 &times; 8.Multiple choice
  3. 3
    What is 6.43 rounded to 1 significant figure?Multiple choice
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Estimations - Practice

Upper Lower Bounds - Practice