At an orbital observatory, Nova and Luna discovered that the space telescope had paused because its calibration system contained an unnecessarily complicated setting. The display showed 2.5(8.4 + 1.6) + 3.7 + 6.3. Nova used the distributive property to rewrite the first part as 2.5(8.4) + 2.5(1.6), which gave 21 + 4. Luna noticed a quicker route: 8.4 + 1.6 = 10, so 2.5(10) = 25. She then used the commutative property to reorder 3.7 + 6.3 and the associative property to group them as 10. Both approaches produced the same calibration value, 35. To verify that no value had changed during simplification, they evaluated the original expression directly and again obtained 35. Nova and Luna entered 35 as the simplified calibration setting. The telescope accepted the value, rotated toward its target, and resumed collecting images of a distant star system.

Challenges

Start
Challenge 1 of 5

Challenges

Start
Challenge 1 of 4
Hint
Compare 2.5(8.4 + 1.6) with 2.5(8.4) + 2.5(1.6) and consider which calculation is simpler.
Luna simplified 2.5(8.4 + 1.6) by adding inside the parentheses first, while Nova used the distributive property. Which statement correctly compares their methods?
Only Nova's method is valid because multiplication must always be done before addition inside parentheses.Only Luna's method is valid because the distributive property cannot be used with decimals.Both methods are valid, but Luna's method is more efficient because 8.4 + 1.6 equals 10.Both methods are valid, but they must produce different values because they use different operations first.
Correct. Both methods preserve the value, but adding 8.4 and 1.6 first makes the calculation especially simple.
Not quite. Both methods are mathematically valid, so compare their efficiency rather than assuming one changes the value.
Next
🎉 Congrats! You got this question!
Hint
Look for pairs of decimals that can be grouped to make 10.
For a second telescope calibration, the display keeps 2.5(8.4 + 1.6) but replaces 3.7 + 6.3 with 4.2 + 5.8. What is the value of the new calibration expression?
Correct. The new calibration expression also has a value of 35.
Check the sums inside the parentheses and in the final pair before combining the results.
Next
🎉 Congrats! You got this question!
Hint
Look for two sums that make 10 while preserving the operations in the original expression.
Which rearrangement of the original calibration expression most directly uses convenient grouping to show that its value is 35?
2.5(8.4) + 1.6 + 3.7 + 6.32.5(8.4 + 1.6 + 3.7 + 6.3)(2.5 + 8.4 + 1.6) + (3.7 + 6.3)2.5(10) + (3.7 + 6.3) = 25 + 10 = 35
Correct. Grouping 8.4 + 1.6 and 3.7 + 6.3 creates convenient sums without changing the expression's value.
Not quite. Make sure the rearrangement preserves the original multiplication and uses convenient groups.
Next
🎉🎉🎉🎉🎉🎉🎉🎉
🎉 Congrats! You finished this challenge set!
Uzay Teleskobunu Yeniden Kalibre Etmek
TR
ARC_G10_NUM_02_STORY_02
Uzay Teleskobunu Yeniden Kalibre Etmek
Recalibrating the Space Telescope
EN
ARC_G10_NUM_02_STORY_02
Recalibrating the Space Telescope