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SL 1.6—Approximating and estimating - IB Questionbank | RevisionDojo

Practice SL 1.6—Approximating and estimating with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

The diameter of Earth is $1.274 \times 10^{7}$ , correct to four significant figures. The circumference of Mars is $2.134 \times 10^{7}$ , correct to four significant figures.

Modelling Earth and Mars as perfect spheres, find the difference between the volume of Earth and the volume of Mars, giving your answer in the form $a \times 10^{k}$ , where $1 \le a < 10$ and $k \in \mathbb{Z}$ .

[5]

Question 2

SLPaper 2

A cylinder has a radius of $12.7$ and a height of $14.4$ .

Calculate the volume of the cylinder, correct to one decimal place.

[2]

Calculate the volume of the cylinder, correct to three significant figures.

[1]

Calculate the volume of the cylinder, correct to the nearest integer.

[1]

Write your answer to part 2 in the form $a \times 10^{k}$ , where $1 \le a < 10$ and $k \in \mathbb{Z}$ .

[1]

Question 3

SLPaper 1

Iron in the asteroid 16 Psyche is said to be valued at $8973$ quadrillion euros (EUR), where one quadrillion $= 10^{15}$.

James believes the asteroid is approximately spherical with radius $113 \text{ km}$. He uses this information to estimate its volume.

Write down the value of the iron in the form $a \times 10^k$ where $1 \le a < 10, k \in \mathbb{Z}$.

[2]

Calculate James’s estimate of its volume, in $\text{km}^3$.

[2]

The actual volume of the asteroid is found to be $6.074 \times 10^6 \text{ km}^3$.

Find the percentage error in James’s estimate of the volume.

[2]

Question 4

SLPaper 2

Lucy found that the exact answer for $S$ is $\frac{45}{16}$

Write down the exact answer as a decimal.

[1]

Lucy rounds her exact answer so that the percentage error is $6.67 \%$ . State the number of significant figures Lucy rounded the exact answer to and write down the approximate value of $S$ used in calculating the percentage error.

[2]

Question 5

SLPaper 2

Let $S=\left(a \sin ^{2} 4 b\right)\left(c^{2} \tan ^{2} 12 d\right)^{-1}(\sqrt{a}+c-\cos 48 b)$ , where $a=16, b=7.5^{\circ}, c=3, d=5^{\circ}$ .

Find the value of $S$ , giving your answer as a fraction.

[3]

Let $X=\frac{\sqrt{a}+c^{2}-2 \sin 54 d}{\sqrt{a^{3}}-a-c}$ . Find the value of $X$ , giving your answer as a fraction.

[3]

John estimates the value of $S X$ to be 0.3. Calculate the percentage error in John's estimate.

[3]

Question 6

SLPaper 2

Let $P = \frac{30 \sin 2a}{8b}$ , where $a = 45^{\circ}$ and $b = 2$ .

Calculate the exact value of $P$ .

[2]

Give your answer from part 1 correct to two significant figures.

[1]

Allen estimates the value of $P$ to be $2$ . Calculate the percentage error in Allen's estimate.

[2]

Question 7

SLPaper 2

Consider the numbers $a=4.14 \times 10^{6}$ and $b=2.54 \times 10^{-7}$

Calculate $C=\sqrt[10]{\left(\frac{a}{b}\right)^{3}}$ . Give your answer correct to the nearest integer.

[2]

Give your answer to part 1 in the form $A \times 10^{k}$ , where $1 \leq A < 10$ and $k \in \mathbb{Z}$ .

[1]

Question 8

SLPaper 2

Let $Q = \frac{4x}{6 \cos 5y}$ , where $x = 1.25$ and $y = 36^{\circ}$ .

(a) Calculate the exact value of $Q$ as a fraction.

[2]

(b) Give your answer to part (a) correct to three significant figures.

[1]

[2]

Question 9

SLPaper 2

A shop sells bags of potatoes labelled as 5 kg. The shop owner weighs five bags at random and records the following weights, in kilograms:

4.96, 4.89, 5.07, 5.11, 5.02

Find the mean weight of the shop owner's recorded weights.

[2]

Calculate the percentage error between the mean weight and the stated weight of 5 kg.

[2]

The supplier also weighs a sample of five bags. The weights of the first four bags are 5.05 kg, 5.01 kg, 4.97 kg, and 5.09 kg. The weight of the fifth bag is $X$ kg.

Find the interval for the values of $X$ such that the percentage error of the mean weight of these five bags, relative to the stated weight, is less than 1%.

[6]

Question 10

SLPaper 2

Calculate the following, giving your answers in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in \mathbb{Z}$

$4 \times \left(6.2 \times 10^{-5}\right)$

[2]

$\left(4 \times 10^{5}\right)-\left(5 \times 10^{4}\right)$

[2]

$(4321)^{-1}\left(1.2 \times 10^{-1}\right)$

[2]

Paper

Level

Question 1

SLPaper 2

The diameter of Earth is $1.274 \times 10^{7}$ , correct to four significant figures. The circumference of Mars is $2.134 \times 10^{7}$ , correct to four significant figures.

[5]

Question 2

SLPaper 2

A cylinder has a radius of $12.7$ and a height of $14.4$ .

Calculate the volume of the cylinder, correct to one decimal place.

[2]

Calculate the volume of the cylinder, correct to three significant figures.

[1]

Calculate the volume of the cylinder, correct to the nearest integer.

[1]

Write your answer to part 2 in the form $a \times 10^{k}$ , where $1 \le a < 10$ and $k \in \mathbb{Z}$ .

[1]

Question 3

SLPaper 1

Iron in the asteroid 16 Psyche is said to be valued at $8973$ quadrillion euros (EUR), where one quadrillion $= 10^{15}$.

James believes the asteroid is approximately spherical with radius $113 \text{ km}$. He uses this information to estimate its volume.

Write down the value of the iron in the form $a \times 10^k$ where $1 \le a < 10, k \in \mathbb{Z}$.

[2]

Calculate James’s estimate of its volume, in $\text{km}^3$.

[2]

The actual volume of the asteroid is found to be $6.074 \times 10^6 \text{ km}^3$.

Find the percentage error in James’s estimate of the volume.

[2]

Question 4

SLPaper 2

Lucy found that the exact answer for $S$ is $\frac{45}{16}$

Write down the exact answer as a decimal.

[1]

[2]

Question 5

SLPaper 2

Let $S=\left(a \sin ^{2} 4 b\right)\left(c^{2} \tan ^{2} 12 d\right)^{-1}(\sqrt{a}+c-\cos 48 b)$ , where $a=16, b=7.5^{\circ}, c=3, d=5^{\circ}$ .

Find the value of $S$ , giving your answer as a fraction.

[3]

Let $X=\frac{\sqrt{a}+c^{2}-2 \sin 54 d}{\sqrt{a^{3}}-a-c}$ . Find the value of $X$ , giving your answer as a fraction.

[3]

John estimates the value of $S X$ to be 0.3. Calculate the percentage error in John's estimate.

[3]

Question 6

SLPaper 2

Let $P = \frac{30 \sin 2a}{8b}$ , where $a = 45^{\circ}$ and $b = 2$ .

Calculate the exact value of $P$ .

[2]

Give your answer from part 1 correct to two significant figures.

[1]

Allen estimates the value of $P$ to be $2$ . Calculate the percentage error in Allen's estimate.

[2]

Question 7

SLPaper 2

Consider the numbers $a=4.14 \times 10^{6}$ and $b=2.54 \times 10^{-7}$

Calculate $C=\sqrt[10]{\left(\frac{a}{b}\right)^{3}}$ . Give your answer correct to the nearest integer.

[2]

Give your answer to part 1 in the form $A \times 10^{k}$ , where $1 \leq A < 10$ and $k \in \mathbb{Z}$ .

[1]

Question 8

SLPaper 2

Let $Q = \frac{4x}{6 \cos 5y}$ , where $x = 1.25$ and $y = 36^{\circ}$ .

(a) Calculate the exact value of $Q$ as a fraction.

[2]

(b) Give your answer to part (a) correct to three significant figures.

[1]

[2]

Question 9

SLPaper 2

A shop sells bags of potatoes labelled as 5 kg. The shop owner weighs five bags at random and records the following weights, in kilograms:

4.96, 4.89, 5.07, 5.11, 5.02

Find the mean weight of the shop owner's recorded weights.

[2]

Calculate the percentage error between the mean weight and the stated weight of 5 kg.

[2]

The supplier also weighs a sample of five bags. The weights of the first four bags are 5.05 kg, 5.01 kg, 4.97 kg, and 5.09 kg. The weight of the fifth bag is $X$ kg.

Find the interval for the values of $X$ such that the percentage error of the mean weight of these five bags, relative to the stated weight, is less than 1%.

[6]

Question 10

SLPaper 2

Calculate the following, giving your answers in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in \mathbb{Z}$

$4 \times \left(6.2 \times 10^{-5}\right)$

[2]

$\left(4 \times 10^{5}\right)-\left(5 \times 10^{4}\right)$

[2]

$(4321)^{-1}\left(1.2 \times 10^{-1}\right)$

[2]

Paper

Level

Question 1

SLPaper 2

The diameter of Earth is $1.274 \times 10^{7}$ , correct to four significant figures. The circumference of Mars is $2.134 \times 10^{7}$ , correct to four significant figures.

[5]

Question 2

SLPaper 2

A cylinder has a radius of $12.7$ and a height of $14.4$ .

Calculate the volume of the cylinder, correct to one decimal place.

[2]

Calculate the volume of the cylinder, correct to three significant figures.

[1]

Calculate the volume of the cylinder, correct to the nearest integer.

[1]

Write your answer to part 2 in the form $a \times 10^{k}$ , where $1 \le a < 10$ and $k \in \mathbb{Z}$ .

[1]

Question 3

SLPaper 1

Iron in the asteroid 16 Psyche is said to be valued at $8973$ quadrillion euros (EUR), where one quadrillion $= 10^{15}$.

James believes the asteroid is approximately spherical with radius $113 \text{ km}$. He uses this information to estimate its volume.

Write down the value of the iron in the form $a \times 10^k$ where $1 \le a < 10, k \in \mathbb{Z}$.

[2]

Calculate James’s estimate of its volume, in $\text{km}^3$.

[2]

The actual volume of the asteroid is found to be $6.074 \times 10^6 \text{ km}^3$.

Find the percentage error in James’s estimate of the volume.

[2]

Question 4

SLPaper 2

Lucy found that the exact answer for $S$ is $\frac{45}{16}$

Write down the exact answer as a decimal.

[1]

[2]

Question 5

SLPaper 2

Let $S=\left(a \sin ^{2} 4 b\right)\left(c^{2} \tan ^{2} 12 d\right)^{-1}(\sqrt{a}+c-\cos 48 b)$ , where $a=16, b=7.5^{\circ}, c=3, d=5^{\circ}$ .

Find the value of $S$ , giving your answer as a fraction.

[3]

Let $X=\frac{\sqrt{a}+c^{2}-2 \sin 54 d}{\sqrt{a^{3}}-a-c}$ . Find the value of $X$ , giving your answer as a fraction.

[3]

John estimates the value of $S X$ to be 0.3. Calculate the percentage error in John's estimate.

[3]

Question 6

SLPaper 2

Let $P = \frac{30 \sin 2a}{8b}$ , where $a = 45^{\circ}$ and $b = 2$ .

Calculate the exact value of $P$ .

[2]

Give your answer from part 1 correct to two significant figures.

[1]

Allen estimates the value of $P$ to be $2$ . Calculate the percentage error in Allen's estimate.

[2]

Question 7

SLPaper 2

Consider the numbers $a=4.14 \times 10^{6}$ and $b=2.54 \times 10^{-7}$

Calculate $C=\sqrt[10]{\left(\frac{a}{b}\right)^{3}}$ . Give your answer correct to the nearest integer.

[2]

Give your answer to part 1 in the form $A \times 10^{k}$ , where $1 \leq A < 10$ and $k \in \mathbb{Z}$ .

[1]

Question 8

SLPaper 2

Let $Q = \frac{4x}{6 \cos 5y}$ , where $x = 1.25$ and $y = 36^{\circ}$ .

(a) Calculate the exact value of $Q$ as a fraction.

[2]

(b) Give your answer to part (a) correct to three significant figures.

[1]

[2]

Question 9

SLPaper 2

A shop sells bags of potatoes labelled as 5 kg. The shop owner weighs five bags at random and records the following weights, in kilograms:

4.96, 4.89, 5.07, 5.11, 5.02

Find the mean weight of the shop owner's recorded weights.

[2]

Calculate the percentage error between the mean weight and the stated weight of 5 kg.

[2]

The supplier also weighs a sample of five bags. The weights of the first four bags are 5.05 kg, 5.01 kg, 4.97 kg, and 5.09 kg. The weight of the fifth bag is $X$ kg.

Find the interval for the values of $X$ such that the percentage error of the mean weight of these five bags, relative to the stated weight, is less than 1%.

[6]

Question 10

SLPaper 2

Calculate the following, giving your answers in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in \mathbb{Z}$

$4 \times \left(6.2 \times 10^{-5}\right)$

[2]

$\left(4 \times 10^{5}\right)-\left(5 \times 10^{4}\right)$

[2]

$(4321)^{-1}\left(1.2 \times 10^{-1}\right)$

[2]

SL 1.6—Approximating and estimating Questionbank