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Problem-solving Pack

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Ecolint Campus des NationsMathematics
Year 7 · 7.2 Directed Numbers

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Diophantine puzzle. Find all pairs of positive integers (x,y)(x, y) that satisfy 3x+5y=473x + 5y = 47.

List every solution. Explain why there are finitely many.

Working space

2Problem 2 of 12
Number line puzzle. Three points A, B, C lie on a number line. A is at −14. B is exactly halfway between A and C. The distance from A to C is 22.

(a) What is the coordinate of C?
(b) What is the coordinate of B?
(c) What is B × A?

Working space

3Problem 3 of 12
Temperature number line investigation. The temperatures (°C) recorded at six cities at midnight are:

Oslo: −18, Moscow: −23, Rome: 4, Cairo: 12, Reykjavik: −7, Seoul: −11.

(a) Write the cities in order from coldest to warmest.
(b) What is the range of temperatures?
(c) A city X has a temperature exactly halfway between Moscow and Cairo. What is X's temperature?
(d) If every temperature rises by 15°C, how many cities are then above 0°C?

Working space

4Problem 4 of 12
Magic square with negatives. Complete the 3 × 3 magic square below so that every row, column, and diagonal has the same sum. Some entries are given.

−5□3□−1□1□−3\begin{array}{|c|c|c|}\hline {-5} & \square & {3} \\\hline \square & {-1} & \square \\\hline {1} & \square & {-3} \\\hline\end{array}

Working space

5Problem 5 of 12
Frog jumping puzzle. Frogs sit on lily pads in a row. Red frogs face right; blue frogs face left. They need to swap sides. The rules:
- A frog can move one space forward onto an empty pad.
- A frog can jump over exactly one frog of the opposite colour onto an empty pad.
- No frog may move backwards.

(a) With 1 red and 1 blue frog (3 pads: R _ B → B _ R), what is the minimum number of moves?
(b) With 2 red and 2 blue frogs (5 pads), what is the minimum number of moves?
(c) With 3 red and 3 blue frogs (7 pads), what is the minimum number of moves?
(d) Spot the pattern and predict the minimum moves for nn red and nn blue frogs.

Working space

6Problem 6 of 12
Sign rules investigation. Without using a calculator, decide whether each expression is positive or negative. Then calculate the exact value.

(a) (−3)×(−4)×(−2)(−3) × (−4) × (−2)
(b) (−2)5(−2)^5
(c) (−1)100(−1)^{100}
(d) (−6)2(−3)\dfrac{(−6)^2}{(−3)}

Working space

7Problem 7 of 12
Factor and sign puzzle. Two integers satisfy all three conditions:
- Their product is +36
- Their sum is negative
- Both integers are negative

List all possible pairs. Which pair has the smallest (most negative) sum?

Working space

8Problem 8 of 12
Prime factorisation and directed numbers. The prime factorisation of a number nn is 23×322^3 \times 3^2.

(a) Write down the value of nn.
(b) Write down all the factors of nn that are greater than 10.
(c) A temperature starts at −n-n°C. It rises by n\sqrt{n}°C. What is the new temperature? (Leave your answer in terms of a square root if nn is not a perfect square.)

Working space

9Problem 9 of 12
Directed number patterns. Here is a pattern of directed numbers:

Row 1: −1−1
Row 2: −2,−1−2, −1
Row 3: −3,−2,−1−3, −2, −1
Row 4: −4,−3,−2,−1−4, −3, −2, −1

(a) What is the sum of the numbers in Row 4?
(b) Find a formula for the sum of Row nn.
(c) For which row is the sum equal to −55-55?

Working space

10Problem 10 of 12
Overdrawn accounts. Three friends have the following bank balances: Alice £−24, Ben £−15, Chris £48.

(a) Who has the least money? Who has the most?
(b) Chris transfers enough money to Alice and Ben so that all three balances are equal. How much does each person end up with?
(c) How much did Chris transfer in total?

Working space

11Problem 11 of 12
Connecting HCF and negative numbers. Two numbers pp and qq satisfy:
- p×q=−180p × q = -180
- HCF(∣p∣,∣q∣)=6\text{HCF}(|p|, |q|) = 6
- p>0p > 0 and q<0q < 0
- ∣p∣<∣q∣|p| < |q|

Find all possible pairs (p,q)(p, q).

Working space

12Problem 12 of 12
Temperature data. A class records the daily high temperature (°C) for one week:

−3, 1, −5, 2, −1, 4, −2−3, \ 1, \ −5, \ 2, \ −1, \ 4, \ −2


(a) What is the range of temperatures?
(b) What is the mean temperature? (Give your answer as a fraction if not exact.)
(c) On how many days was the temperature below the mean?
(d) The following week every temperature is double the previous week's. Does the mean double? Does the range double? Explain.

Working space