MYP 3 · Maths
REAL NUMBERS AND RATIO
Decimal numbers
QUESTION 1 [7 marks] — Criterion B
Medium
Investigate the decimal expansions of sevenths: $\frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \ldots, \frac{6}{7}$.
a.
Using a calculator, write out $\frac{1}{7}$ and $\frac{2}{7}$ as decimals, to at least 6 decimal places.
[2] Show Solution
$\frac{1}{7} = 0.142857142857\ldots$ (repeating). $\frac{2}{7} = 0.285714285714\ldots$ (repeating).
b.
Both decimals repeat using the same 6 digits: 1, 4, 2, 8, 5, 7 — just starting at a different point in the cycle. Verify this is also true for $\frac{3}{7}$ and $\frac{4}{7}$.
[3] Show Solution
$\frac{3}{7}=0.428571\ldots$ and $\frac{4}{7}=0.571428\ldots$ — both use the same digit cycle $142857$, confirming the pattern.
c.
Based on this pattern, predict the repeating decimal for $\frac{5}{7}$ without a calculator, then check your answer.
[2] Show Solution
Prediction (continuing the cyclic pattern): $0.714285\ldots$. Checking: $\frac{5}{7}=0.714285714285\ldots$ — matches the prediction.
QUESTION 2 [6 marks] — Criterion C
Medium
A student needs to convert the recurring decimal $0.\dot{4}\dot{5}$ (meaning $0.454545\ldots$) into a fraction.
a.
Let $x = 0.\dot{4}\dot{5}$. Explain why multiplying both sides by 100 is the correct first step (rather than multiplying by 10).
[2] Show Solution
The repeating block has 2 digits ("45"), so multiplying by $10^2=100$ shifts the decimal point exactly one full repeating cycle, meaning the decimal part after the shift lines up exactly with the original — this is essential for the subtraction step to eliminate the repeating part.
b.
Complete the conversion, showing every step clearly: set up $100x$, subtract $x$, and solve for $x$ as a fraction in simplest form.
[4] Show Solution
$100x = 45.454545\ldots$. Subtracting: $100x - x = 45.4545\ldots - 0.4545\ldots$, so $99x = 45$. Then $x = \frac{45}{99} = \frac{5}{11}$ (dividing by HCF 9).
QUESTION 3 [5 marks] — Criterion D
Medium
A currency exchange booth converts Australian dollars (AUD) to Euros (EUR) at a rate of 1 AUD $=$ 0.61 EUR, and charges a flat \$5 AUD service fee, taken before conversion.
a.
A tourist wants to exchange \$350 AUD. Calculate how many Euros they will receive.
[3] Show Solution
After fee: $350 - 5 = 345$ AUD. Converting: $345 \times 0.61 = 210.45$ EUR.
b.
The tourist was expecting approximately 213 EUR (based on the full \$350 without the fee). Explain the difference between their expectation and the actual amount.
[2] Show Solution
Without the fee, $350\times0.61=213.50$ EUR — close to the tourist's expectation. The actual amount (210.45 EUR) is lower because the flat \$5 service fee is subtracted before the exchange rate is applied, reducing the amount actually converted.