Real-World Applications പ്രയോഗങ്ങൾ
Absolute value, distance and midpoint are not just number-line games. Every time you ask "how far apart?", "how far from the target?" or "where is halfway?", you are using this chapter. Move the sliders in each lab below.
- Distance from 0: |x| (കേവലമൂല്യം) – page 6.
- Distance between x and y: |x − y| = bigger − smaller (അകലം) – page 8.
- Halfway point: ½(x + y) (മധ്യബിന്ദു) – page 10.
1. How much colder? Temperatures of two cities
In winter Leh in Ladakh can be −8 °C while Kochi is 30 °C. The difference is the distance between the two numbers on a thermometer: |30 − (−8)| = 38 degrees.
In the exam this looks like: "The temperature in Leh is −8 °C and in Kochi 30 °C. How many degrees warmer is Kochi?" → |30 − (−8)| = 38.
2. Above and below sea level
Heights are measured from sea level (0). Parts of Kuttanad lie about 2 m below sea level (−2 m), while Munnar is about 1600 m above. The difference in height is |1600 − (−2)| = 1602 m.
In the exam this looks like: "A diver is at −35 m and a bird at 20 m above the sea. How far apart are they?" → 20 − (−35) = 55 m.
3. A bank balance that goes below zero
A balance of −₹800 means the account owes ₹800 (an overdraft). After a deposit, the new balance is the old one plus the deposit; how much is owed or saved is the absolute value.
In the exam this looks like: "Balance −₹800; ₹1500 is deposited. What is the new balance?" → −800 + 1500 = ₹700.
4. Minutes before and after kickoff
Call the kickoff time 0. Ten minutes before is −10, a goal in the 25th minute is +25. How far apart are two moments? The distance |t₁ − t₂|.
In the exam this looks like: "The gates open 45 minutes before kickoff and the first goal comes 20 minutes after. How long between them?" → |20 − (−45)| = 65 minutes.
5. Engineering tolerance: |x − 10| ≤ 0.2
A factory makes bolts that should be 10 mm long, "± 0.2 mm". A bolt of length x passes if its distance from 10 is at most 0.2: |x − 10| ≤ 0.2, that is 9.8 ≤ x ≤ 10.2.
In the exam this looks like: "Lengths x with |x − 10| ≤ 0.2 are accepted. What are the smallest and largest accepted lengths?" → 9.8 mm and 10.2 mm.
6. A new bus stop exactly halfway
Two stops are at kilometre 3 and kilometre 11 along a road. A new stop exactly halfway is at their midpoint, ½(3 + 11) = 7. If kilometres to the west of the depot are negative, the same rule works.
In the exam this looks like: "Two stops are at −4 km and 10 km from the depot. Where is the halfway stop?" → ½(−4 + 10) = 3 km.
7. Meeting halfway
Two friends live on the same road at positions a and b. If they walk towards each other at the same speed, they meet at the midpoint – and each walks exactly half the distance between their homes.
In the exam this looks like: "Homes at −5 km and 6 km. Where do they meet, and how far does each walk?" → meet at ½ km; each walks 5½ km.
8. Measurement error
A pencil is truly 25.0 cm long. A student measures 24.6 cm. The error is the distance between the measured and true values, |24.6 − 25.0| = 0.4 cm – it does not matter whether the reading was too big or too small.
In the exam this looks like: "The true value is 25 cm; the readings are 24.6 cm and 25.3 cm. Which is more accurate?" → errors 0.4 and 0.3, so 25.3 cm.
9. The average of two marks is their midpoint
Marks 36 and 44 have average ½(36 + 44) = 40, exactly halfway between them on the number line. The average is always as far from the lower mark as from the higher one.
In the exam this looks like: "The midpoint of two marks is 40 and one mark is 36. What is the other?" → 2 × 40 − 36 = 44.
10. Try these word problems
- "How far apart?" – temperatures, heights, times, balances – is always |x − y|, the bigger minus the smaller.
- "How far from the target?" – tolerances and errors – is |x − target|; "within r" means |x − target| ≤ r.
- "Where is halfway?" – bus stops, meeting points, averages – is ½(x + y).
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