Bounds: Sums and Differences

GCSE Number bounds error intervals
Practice this formula All formulas
\( L_{x+y}=L_x+L_y,\ U_{x+y}=U_x+U_y;\quad L_{x-y}=L_x-U_y,\ U_{x-y}=U_x-L_y \)
ChatGPT said:

Statement

When we add or subtract quantities that have bounds, we can find the new interval by combining the extremes of the input intervals. If \(x \in [L_x, U_x]\) and \(y \in [L_y, U_y]\), then:

\[ L_{x+y} = L_x + L_y, \quad U_{x+y} = U_x + U_y \] \[ L_{x-y} = L_x - U_y, \quad U_{x-y} = U_x - L_y \]

That is, the smallest possible sum comes from adding the two lower bounds; the largest comes from adding the two upper bounds. For differences, the smallest possible result comes from subtracting the largest possible \(y\) from the smallest possible \(x\); the largest difference comes from subtracting the smallest possible \(y\) from the largest possible \(x\).

Why it’s true (short reason)

  • Every possible value of \(x\) lies between \(L_x\) and \(U_x\). Every possible value of \(y\) lies between \(L_y\) and \(U_y\).
  • For sums: Adding the two smallest possible values gives the minimum sum. Adding the two largest gives the maximum sum.
  • For differences: To make \(x-y\) as small as possible, combine the smallest \(x\) with the largest \(y\). To make it as large as possible, combine the largest \(x\) with the smallest \(y\).
  • This works because addition and subtraction are monotonic in each variable (if one goes up, the result goes up or down in a predictable way).

Recipe (how to use it)

  1. Convert any rounded values into intervals \([L,U)\) using half-step rules for nearest unit, decimal places, or significant figures.
  2. For a sum:
    • Lower bound = \(L_x + L_y\)
    • Upper bound = \(U_x + U_y\)
  3. For a difference:
    • Lower bound = \(L_x - U_y\)
    • Upper bound = \(U_x - L_y\)
  4. Write the result as an interval, including correct units if present.

Spotting it

Look for these when:

  • Questions ask for “maximum possible total” or “minimum possible total”.
  • You are subtracting two rounded measurements (e.g. heights, times, lengths).
  • You need to combine measured values (e.g. perimeter of a shape, journey times).

Common pairings

  • Rounding to bounds: before adding/subtracting, always set up the intervals.
  • Geometry problems: adding lengths for perimeters, subtracting inner dimensions (holes, borders).
  • Time intervals: summing times of journeys, subtracting start/end times.

Mini examples

  1. \(x = 12\) (nearest unit), \(y = 7\) (nearest unit). \(x \in [11.5,12.5)\), \(y \in [6.5,7.5)\). \(x+y \in [18,20)\).
  2. \(x = 24.0\) (1 d.p.), \(y = 18.0\) (1 d.p.). \(x \in [23.95,24.05)\), \(y \in [17.95,18.05)\). \(x-y \in [23.95-18.05,\,24.05-17.95] = [5.90,6.10]\).
  3. \(x = 50\) (nearest 10), \(y = 30\) (nearest 10). \(x \in [45,55)\), \(y \in [25,35)\). \(x+y \in [70,90)\).

Pitfalls

  • Mixing endpoints incorrectly: For differences, don’t take lower minus lower or upper minus upper; use the correct pairing (Lx − Uy and Ux − Ly).
  • Forgetting intervals: Never just use the rounded numbers; always expand into bounds first.
  • Units: Make sure both numbers are in the same unit before combining.
  • Signs: If negative values are possible, be extra careful—the rules above assume standard GCSE contexts with positive measurements.

Exam strategy

  • Underline the phrase “nearest…” or “… d.p.” and immediately write the interval for each variable.
  • For sums: add low with low and high with high. For differences: subtract high from low and low from high.
  • Write both ends of the interval and clearly state units.
  • Check that your interval encloses the “obvious” rounded answer (sanity check).

Extended micro-examples

  1. \(x=3.2\) (1 d.p.), \(y=1.5\) (1 d.p.). \(x \in [3.15,3.25)\), \(y \in [1.45,1.55)\). \(x-y \in [3.15-1.55,\,3.25-1.45] = [1.60,1.80]\).
  2. \(x=120\) (nearest 10), \(y=95\) (nearest 5). \(x \in [115,125)\), \(y \in [92.5,97.5)\). \(x-y \in [115-97.5,\,125-92.5) = [17.5,32.5)\).
  3. \(x=15.0\) (1 d.p.), \(y=9.6\) (1 d.p.). \(x \in [14.95,15.05)\), \(y \in [9.55,9.65)\). \(x+y \in [24.50,24.70)\).

Summary

To find bounds for a sum or difference, always start with the measurement intervals. For sums, add the lowest values to get the lower bound and the highest values for the upper bound. For differences, combine the smallest possible \(x\) with the largest possible \(y\) for the minimum, and the largest \(x\) with the smallest \(y\) for the maximum. This guarantees a correct interval that captures every possible result.